Brzostowski–Krasiński–Oleksik conjecture on the Łojasiewicz exponent

Let $f\in\mathbb{C}\{x_0,\ldots,x_n\}$ be Newton nondegenerate, with Newton diagram $\Gamma(f)$. A facet $F\subset\Gamma(f)$ is nonexceptional if it is not an exceptional facet, where an exceptional facet is one for which there exist $i\neq j$ such that all but one vertex lies on the $j$-th coordinate hyperplane and the remaining vertex corresponds to a monomial $x_jx_i^k$. Write $\mathcal{M}(F)$ for the maximal axial number of $F$.

Brzostowski–Krasiński–Oleksik conjecture. If $\Gamma(f)$ contains a nonexceptional facet, then

$$\mathop{\mathcal{L}}(f,0)=\max\left\{\mathcal{M}(F)-1\in\mathbb{Q}\,\middle\vert\,F\text{ is a nonexceptional facet of }\Gamma(f),\ \dim F=n\right\}.$$

The conjecture concerns expressing the Łojasiewicz exponent of a Newton nondegenerate isolated hypersurface singularity purely through the Newton diagram. It was proved in the case $n=2$, but the paper gives a counterexample in the case $n=3$; consequently, the conjecture is refuted in its stated generality.