Combinatorial triangulation conjecture for the Łojasiewicz exponent

Let $f\in\mathbb{C}\{x_0,x_1,\ldots,x_n\}$ be a Newton nondegenerate function germ with an isolated singularity at $0$. Assume that $f$ does not have a Morse point at $0$, or that $n$ is even. Let $\mathcal{T}$ be a triangulation of $\Gamma(f)$, let $\mathcal{T}_{\mathrm{ne}}$ be the set of simplices satisfying

$$\mathop{\mathrm{Cap}}(T)\mathop{\mathrm{CN}_{\mathfrak{N}}}(\mathcal{T}/T)=0,$$

and let $\mathcal{F}_{\mathrm{ne}}^{\mathcal{T}}$ be the associated set of coordinate facets. Then

Combinatorial triangulation conjecture.

$$\mathop{\mathcal{L}}(f,0)=\max_{T\in\mathcal{T}_{\mathrm{ne}}}\deg\bigl(\mathop{\mathrm{CN}_{\mathfrak{N}}}(\mathcal{T}/T)\bigr)-1 =\max_{F\in\mathcal{F}_{\mathrm{ne}}^{\mathcal{T}}}\mathcal{M}(F)-1.$$

This is the paper's fuller triangulation-based proposal, relating the Łojasiewicz exponent to combinatorial Newton data. It is presented after the counterexample to the earlier conjecture, and the source gives no resolution.