Broken graph when `sphinx.ext.inheritance_diagram` `:parts:` directive removes distinguishing path components

Describe the bug

When generating an inheritance diagram using sphinx.ext.inheritance_diagram with a :parts: directive that removes distinguishing components from the path (e.g. with :parts: 2 for the inheritance diagram of class a.b.c.d with base class b.c.d), an inheritance graph is generated consisting of a single node with an edge looping back to itself.

This appears to occur because the base name in the class information constructed by InheritanceGraph._class_info is the partial name that depends on the :parts: directive (as generated by InheritanceGraph.class_name, which, contrary to the claim in its docstring, does not return the fully-qualified name), so there is no ability to disambiguate between multiple classes sharing the same partial name when specifying edges in the graph.

How to Reproduce

Relevant conf.py is rather complex, and the considerable effort that would be required to construct a minimal example does not seem to be necessary since the cause of the bug can be determined by inspection of the implementation of sphinx.ext.inheritance_diagram.InheritanceGraph.

Environment Information

Platform:              linux; (Linux-5.19.0-35-generic-x86_64-with-glibc2.35)
Python version:        3.9.15 | packaged by conda-forge | (main, Nov 22 2022, 15:55:03) 
[GCC 10.4.0])
Python implementation: CPython
Sphinx version:        6.1.3
Docutils version:      0.19
Jinja2 version:        3.1.2
Pygments version:      2.14.0

Sphinx extensions

sphinx.ext.inheritance_diagram

Additional context

No response

Start by reading sphinx.ext.inheritance_diagram.InheritanceGraph, especially _class_info and class_name, and trace how the :parts: value affects class names and graph edges. Reproduce the a.b.c.d and b.c.d case with :parts: 2 if possible, then verify that the diagram retains distinct nodes and does not create a self-loop.