The associated-prime conjecture for initial ideals of lattice ideals

Let $S$ be a polynomial ring, let $I_{\mathcal L}$ be a lattice ideal in $S$, and let $M$ be its initial ideal with respect to some term order. For a prime ideal $P$ in $S$, write $\operatorname{codim}(P)$ for its codimension, and write $\operatorname{proj-dim}_S(S/M)$ for the projective dimension of $S/M$ over $S$. Associated-prime conjecture. There is an associated prime $P \in \operatorname{Ass}(S/M)$ such that

$$\operatorname{codim}(P)=\operatorname{proj-dim}_S(S/M).$$

The preceding result of Hoşten and Thomas shows that initial ideals of lattice ideals satisfy a chain condition on associated primes, but the authors state that it is unknown whether the corresponding conclusion about an associated prime of maximal codimension holds as well.