[BUG]: Home .local libraries missing from sys.path with embed

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

tags/v2.10.4

Problem description

the standard procedure to embed python in c++ using pybind lead to missing home libraries from sys.path.

Reproducible example code

CMakeLists.txt :

cmake_minimum_required(VERSION 3.1)

include(FetchContent)

add_executable(main ./main.cpp)

FetchContent_Declare(
  pybind11
  GIT_REPOSITORY https://github.com/pybind/pybind11.git
  GIT_TAG        v2.10.4
)

FetchContent_GetProperties(pybind11)
if(NOT pybind11_POPULATED)
  FetchContent_Populate(pybind11)
  add_subdirectory(./build/_deps/pybind11-src)
endif()

target_link_libraries(main pybind11::embed)

main.cpp

#include <pybind11/stl.h>
#include <pybind11/pybind11.h>
#include <pybind11/embed.h>
#include <pybind11/numpy.h>

namespace py = pybind11;

int main()
{
    py::scoped_interpreter guard{};

    py::exec("import sys\nprint(sys.path)");
}

running gives :

> cmake -B build .
> cd build
> make
> ./main 
['', '/usr/lib/python310.zip', '/usr/lib/python3.10', '/usr/lib/python3.10/lib-dynload', '/usr/lib/python3.10/site-packages']

in normal python :

In [1]: import sys

In [2]: print(sys.path)
['/usr/bin', '/usr/lib/python310.zip', '/usr/lib/python3.10', '/usr/lib/python3.10/lib-dynload', '', '/home/xxx/.local/lib/python3.10/site-packages', '/usr/lib/python3.10/site-packages']

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

Not a regression

Reproduce the issue with CMakeLists.txt and main.cpp, starting at py::scoped_interpreter and the embedded Python setup. Compare the printed sys.path with normal Python behavior, then verify that the user's home .local site-packages path is included for the embedded interpreter.