Combinatorial interpretation of local coefficients for del Pezzo duals

Construct a natural weighted combinatorial interpretation, related to quadratic triangulations, for the local coefficients $\tau_{\mathbf x}(I)$ arising from the polar duals $V_{2m}^{\vee}$ of del Pezzo polytopes.

Background

Theorem 5.1 establishes strict positivity of the local coefficients $\tau_{\mathbf x}(I)$ in a local decomposition of the magic coefficients. For the arithmetic-node specializations associated with del Pezzo polytopes and cycles, the authors observe that the remaining issue is to understand these positive quantities combinatorially.

The question specifically asks whether the coefficients arising from $V_{2m}^{\vee}$ can be interpreted through a natural weighted combinatorial structure connected with the quadratic triangulations of these polytopes.