Hirsch conjecture for dual graphs of polytope boundary ideals

Let $\Delta$ be the boundary complex of a convex polytope. For its Stanley–Reisner ideal $I_\Delta$, define $G(I_\Delta)$ to be the dual graph of the minimal primes of $I_\Delta$, and call an ideal $I$ Hirsch when

$$\operatorname{diam} G(I)\leq \operatorname{height} I.$$

Hirsch conjecture. If $\Delta$ is the boundary of a convex polytope, then $I_\Delta$ is Hirsch.

This is the algebraic formulation of the Hirsch bound for the graph diameter of the facets of a convex polytope. The source notes that the classical Hirsch problem was solved negatively by Santos, so this conjecture concerns the corresponding bound for the dual graph of the Stanley–Reisner ideal.