Definable triangulation conjecture for functions on compact definable sets

Let $f:K\to\mathbb{R}$ be a definable function on a compact definable set $K\subset\mathbb{R}^m$. A definable triangulation of $K$ is a definable triangulation whose open simplices satisfy the following conditions. For each $n\leq\dim K$ and each open $n$-simplex $\Delta$ of the triangulation, let

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

be the graph of the restriction of $f$ to $\Delta$. Definable triangulation conjecture. There exists a definable triangulation of $K$ such that $\Gamma$ is a topologically regular $n$-cell, and either $f$ is constant on $\Delta$ or every nonempty level set

$$\Gamma\cap\{t=\text{constant}\}$$

is a topologically regular $(n-1)$-cell. This conjecture was proposed as motivation for studying semi-monotone sets, monotone functions, and monotone maps; the paper develops those theories and proves related regular-cell results, but the supplied text does not state whether this triangulation conjecture itself is resolved.