Eliahou–Kryuchkov signed diagonal flip conjecture for polygon triangulations

Let $P$ be a polygon, and let a signed triangulation of $P$ mean a triangulation whose triangles are assigned signs. A signed diagonal flip replaces the diagonal of a quadrilateral formed by two adjacent triangles when the move is allowed by their signs, assigning the opposite signs to the two resulting triangles.

Eliahou–Kryuchkov conjecture. Given two triangulations of the same polygon, there exist signs for them such that it is possible to transform one to the other by a finite sequence of signed diagonal flips.

Signed diagonal flips preserve the vertex coloring associated with the Four Color Theorem, so this conjecture would imply the theorem. The source presents it as conjectured by Eliahou and Kryuchkov; no resolution is supplied here.