Multiple parsing files with menhir

Question

I'm trying to have two different parsers with different entry points.

I have the following project hierarchy:

.
├── bin
│   ├── dune
│   ├── lexer.mll
│   ├── main.ml
│   ├── parser.messages
│   ├── parser.mly
│   └── test_parser.mly
├── dune-project
├── program.opam
└── test

lexer.mll

{
  open Parser
  exception Error of string
}

rule token = parse
| [' ' '\t'] { token lexbuf }
| '\n' { EOL }
| ['0'-'9']+ as i { INT (int_of_string i) }
| '+' { PLUS }
| eof { exit 0 }
| _   { raise Exit }

parser.mly

%token <int> INT
%token PLUS
%token EOL

%left PLUS

%start <int> main

%%

main:
| e = expr EOL
    { e }

%public expr:
| i = INT { i }
| e1 = expr PLUS e2 = expr { e1 + e2 }

and dune

(ocamllex
 (modules lexer))

(menhir
 (modules parser)
 (flags --table)
)

(executable
 (name main)
 (public_name program)
 (libraries menhirLib)
)

(rule
 (targets parser_messages.ml)
 (deps parser.messages parser.mly)
 (action  (with-stdout-to %{targets} (run menhir --compile-errors %{deps}))))

This works perfectly (the parser.messages has been generated by menhir --list-errors bin/parser.mly > bin/parser.messages and is not really important)

However, I'd like to have a second parser that uses some rules defined in parser.mly and according to http://gallium.inria.fr/~scherer/tmp/menhir-manual/main.html#sec23 it should be possible but I guess I'm doing it wrong:

test_parser.mly

%start <int> test

%%

test:
| el = separated_nonempty_list(COMMA, expr) EOL
    { List.fold_left (+) 0 e }

And in my dune file I added:

(menhir
 (modules parser test_parser)
 (merge_into test_parser)
 (flags --table))


(rule
 (targets test_parser_messages.ml)
 (deps test_parser.messages test_parser.mly)
 (action  (with-stdout-to %{targets} (run menhir --compile-errors %{deps}))))

But when I dune build:

❯ dune build
File "test_parser.mly", line 6, characters 21-25:
Error: expr is undefined.
File "bin/test_parser.mly", line 7, characters 27-28:
Error: Unbound value e

expr is preceded by %public in parser.mly so I thought I could use it from test_parser.mly. Can this feature only be used with a common entry point and different producing rules?