Gabrielov's regular triangulation conjecture for definable functions

Let $K e\varnothing$ be a compact definable set in ${\mathbb R}^m$, and let $f:K\to{\mathbb R}$ be a definable function. Write the graph over an open simplex $\Delta$ as

$$\Gamma=\{(\mathbf{x},t)\mid \mathbf{x}\in\Delta,\ t=f(\mathbf{x})\}.$$

A regular $n$-cell is understood as in the source's Definition 1.1. A definable triangulation of $K$ is a triangulation whose simplices are definable.

Gabrielov's regular triangulation conjecture. There exists a definable triangulation of $K$ such that, for each $n\leq\dim K$ and every open $n$-simplex $\Delta$ of the triangulation, $\Gamma$ is a regular $n$-cell, and either $f$ is constant on $\Delta$ or every nonempty level set

$$\Gamma\cap\{t=c\}$$

is a regular $(n-1)$-cell.

This conjecture is presented as a key step in Gabrielov's triangulation program: a compatible triangulation of increasing definable families of compact sets would imply the homotopy equivalence conjecture above. The source does not provide a resolution, so the conjecture remains open.