When having multiple root toctrees, how to have them correspond to a part in latex output?

[PierreMarchand20]

I have several toctree like this:

.. toctree::
   :caption: My first part

   chapter_1_of_first_part
   chapter_2_of_first_part

.. toctree::
   :caption: My second part

   chapter_1_of_second_part
   chapter_2_of_second_part

.. toctree::
   :caption: My third part

   chapter_1_of_third_part
   chapter_2_of_third_part

Is it possible to tell sphinx to define a part for each toctree, and use chapter within each toctree for numbering for latex output ?

At the moment, it gives me an output with a table of content in latex whose name is "My first part", and all chapters together.

Here is the real life example : https://github.com/htool-ddm/htool_documentation/blob/update_htool/docs/source/index.rst and the associated output : https://htool-ddm.pages.math.cnrs.fr/htool-ddm.pdf.