GHV upper-bound conjecture for Stanley–Reisner ideals of matroids

Let $M$ be a matroid and let $I_{\Delta(M)}$ be its Stanley–Reisner ideal. Write $\widehat{\rho}(I_{\Delta(M)})$ for its asymptotic resurgence, $\omega(I_{\Delta(M)})$ for the largest degree of a minimal generator, and $\widehat{\alpha}(I_{\Delta(M)})$ for its Waldschmidt constant. GHV upper-bound conjecture.

$$\widehat{\rho}(I_{\Delta(M)})\le\frac{\omega(I_{\Delta(M)})}{\widehat{\alpha}(I_{\Delta(M)})}.$$

For matroid Stanley–Reisner ideals, $\omega(I_{\Delta(M)})$ is the largest size of a circuit of $M$. The bound is known for ideals of smooth subschemes and has been checked in the source for all simple matroids on ground sets of size at most $8$, but is conjectured in general.