Andrews–Curtis conjecture for contractible simple polyhedra

A simple polyhedron is a compact $2$-dimensional polyhedron whose points have neighborhoods modeled on the standard local models for simple polyhedra. A gleam is the extra half-integer framing data assigned to regions of a shadow; here gleams are ignored. The pentagon, hexagon, and $2$-$0$ moves are the local moves on simple polyhedra described in the source, and their inverses are also allowed.

Andrews–Curtis conjecture. Any two contractible simple polyhedra can be related by a sequence of pentagon, hexagon, and $2$-$0$ moves and their inverses, ignoring all gleams.

This is presented as a well-known conjecture closely related to the shadow-calculus question for contractible shadows. The supplied text does not state a resolution, so its status is left open.