[BUG]: Incompatible function arguments when using py::function with default parameter

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What version (or hash if on master) of pybind11 are you using?

2.11.1

Problem description

When setting up a function that takes a py::function as an input argument AND which lets that argument default to nullptr or py::none(), that function cannot be used from Python with default arguments.

The workaround is to use py::object and just work with it as if it were a py::function.

A similar issue was reported previously in https://github.com/pybind/pybind11/issues/3205, but for some reason that issue was closed. It's worth noting that in my code example I set the default argument to nullptr but we see the same behavior if the default argument is py::none(). If the default argument is py::object() or py::function() then you get the issue reported in #3205, namely that we cannot even import the function because we get the error "cannot convert default argument in func2 into a Python object (type not registered yet?)"

Reproducible example code

#include <pybind11/pybind11.h>

namespace py = pybind11;
using namespace pybind11::literals;

PYBIND11_MODULE(example, m) {
  m.def("func1", [](py::object   func_to_call){}, "docstring", "func_to_call"_a=nullptr);
  m.def("func2", [](py::function func_to_call){}, "docstring", "func_to_call"_a=nullptr);
  /*Python:
from example import func1, func2

func1(lambda x: x)  # works
func1()  # works
func2(lambda x: x)  # works
func2()  # incompatible argument error
*/
}

Is this a regression? Put the last known working version here if it is.

Not a regression

Start with the pybind11/pybind11.h binding entry point and the provided PYBIND11_MODULE example, comparing func1's py::object default handling with func2's py::function handling. Reproduce the incompatible-argument error for func2() and verify that the default nullptr or py::none() is accepted without changing the working behavior of func1().