Printing to multiple printers from .NET WebBrowser - program hangs unless a MessageBox is used

Question

The WebBrowser.Print method has the limitation of not allowing the caller to specify a printer other than the system's default one. As a workaround, it has been suggested^{[1], [2]} to alter the system's default printer prior to calling Print(), however it's also reported^[3] (and I experienced firsthand) that the WebBrowser instance will continue to print to the previously defined printer even after the system default is altered.

To work around that, registering a handler to the PrintTemplateTeardown event by accessing the underlying ActiveX object of the managed WebBrowser instance and waiting for the event to fire before printing further documents has been proposed^{[4], [5]}, and that is what I am trying to implement. I simplified what is a much more complex program to the MVCE presented below.

(The program is a .NET Core 3.1 Windows Forms application, with one form containing nothing more than a BackgroundWorker object named bw.)

Form1.cs

using System;
using System.ComponentModel;
using System.Linq;
using System.Threading.Tasks;
using System.Windows.Forms;
using System.Threading;
using System.Drawing.Printing;
using System.Management;

namespace Demo_1
{
    public partial class Form1 : Form
    {
        public Form1()
        {
            InitializeComponent();

            bw.RunWorkerAsync();
        }

        private void bw_DoWork(object sender, DoWorkEventArgs e)
        {
            while (true)
            {
                BwPolling();

                Thread.Sleep(20000);
            }
        }

        private void BwPolling()
        {
            string[] htmlStrings = { "test1", "test2" };

            foreach (string html in htmlStrings)
            {
                Invoke((MethodInvoker)delegate
                {
                    PrinterSettings.StringCollection installedPrinters = PrinterSettings.InstalledPrinters;
                    foreach (string printer in installedPrinters)
                    {
                        string[] validPrinterNames =
                        {
                            "Microsoft Print to PDF",
                            "Microsoft XPS Document Writer"
                        };

                        if ( validPrinterNames.Contains(printer) )
                        {
                            SetDefaultPrinter(printer);
                            
                            var wb = new WebBrowser();
                            wb.DocumentText = html;

                            // With inspiration from code by Andrew Nosenko <https://stackoverflow.com/users/1768303/noseratio>
                            // From: https://stackoverflow.com/a/19737374/3258851
                            // CC BY-SA 3.0

                            var wbax = (SHDocVw.WebBrowser)wb.ActiveXInstance;

                            TaskCompletionSource<bool> printedTcs = null;
                            SHDocVw.DWebBrowserEvents2_PrintTemplateTeardownEventHandler printTemplateTeardownHandler =
                            (p)
                                => printedTcs.TrySetResult(true); // turn event into awaitable task

                            printedTcs = new TaskCompletionSource<bool>();
                            wbax.PrintTemplateTeardown += printTemplateTeardownHandler;

                            try
                            {
                                MessageBox.Show("Printing to " + printer);

                                wb.Print();

                                printedTcs.Task.Wait();
                            }

                            finally
                            {
                                wbax.PrintTemplateTeardown -= printTemplateTeardownHandler;
                            }

                            wb.Dispose();
                        }
                    }
                });
            }
        }

        private static bool SetDefaultPrinter(string name)
        {
            // With credits to Austin Salonen <https://stackoverflow.com/users/4068/austin-salonen>
            // From: https://stackoverflow.com/a/714543/3258851
            // CC BY-SA 3.0
            using ( ManagementObjectSearcher objectSearcher = new ManagementObjectSearcher("SELECT * FROM Win32_Printer") )
            {
                using ( ManagementObjectCollection objectCollection = objectSearcher.Get() )
                {
                    foreach (ManagementObject mo in objectCollection)
                    {
                        if ( string.Compare(mo["Name"].ToString(), name, true) == 0 )
                        {
                            mo.InvokeMethod("SetDefaultPrinter", null);
                            return true;
                        }
                    }
                }
            }
            return false;
        }
    }
}

The problem being faced is when I remove the message box from the BwPolling() method, right before calling Print(), i.e. when this line is removed:

MessageBox.Show("Printing to " + printer);

then the program freezes, nothing is printed, and the process must eventually be terminated.

I believe I can sort of understand the issue on its surface: WebBrowser requires an STA thread with an active message loop^{[6], [7]}; by calling printedTcs.Task.Wait(); within a Invoke((MethodInvoker)delegate block (called on the Form1 instance; this. is ommited), I am blocking the STA thread and the application hangs waiting for an event that is never fired. This is in fact mentioned in a comment under the answer I credited in my code.

Just can't figure out what a proper solution would be. Got lost in attempts to run the printing routine in a secondary thread. Maybe something wrong in my execution, guess I require assistance in this. Any help?

Thanks.