Google tag manager data layer within iframes

Question

I am no sure how to setup the dataLayer across iframes with the Google tag manager.
I found the following information: Google Tag Manager dataLayer within iframes

On the website is the iFrame. The iFrame is loaded on an sudomain.

Parentpage: www.domain.com

iFrame: www.sudomain.domain.com

Is it correct to setup the GTM container on the parentpage and define all dataLayer variable there and push the dataLayer from the iFrame via the parent.dataLayer.push method?

Does the parent.dataLayer.push method then push the dataLayer values automatically to the dataLayer on the parentpage?

For example:

Parentpage

dataLayer = [{
        'country': 'DE',
        'pageType': 'product'
    }]

Then the iFrame will be loaded with the parent.dataLayer.push method:

parent.dataLayer.push({
        'country': 'DE',
        'pageType': 'product123'    });

Will the dataLayer value for 'pageType' in the parentpage then automatically be overwritten by the iframe value)?

Thanks