The two-dimensional Weil conjecture for lattice partition functions

Let $X$ be an object of the tensor category $\mathcal C_{\mathbb F_q}$, let $M\in\operatorname{End}_{\mathcal C_{\mathbb F_q}}(X)$, and let $Z_M(\Lambda)$ be the lattice function defined for finite-index lattices in $\mathbb Z^2$ by the numerical realizations and Frobenius operators. Let $Z_R^{lat}(\Lambda)$ denote the partition function of super Boltzmann data $(V_1,V_2,R)$.

Two-dimensional Weil conjecture. For every endomorphism $M\in\operatorname{End}_{\mathcal C_{\mathbb F_q}}(X)$ there exist super Boltzmann data $(V_1,V_2,R)$ such that, for every finite-index lattice $\Lambda\subset\mathbb Z^2$,

$$Z_M(\Lambda)=Z_R^{lat}(\Lambda).$$

This proposes a lattice-model realization of the two-dimensional motivic partition function, extending the preceding rationality result for $Z_M$. The source gives no proof of this realization.