Equality of hypergraph and graph zero-free loci

For a positive integer $\Delta$, let $\mathcal U_\Delta$ be the maximal simply connected open subset of $\mathbb C$ containing $0$ that is zero-free for the independence polynomials of all graphs of maximum degree $\Delta$. Let $\mathcal U_{\Delta,k}$ be the analogous set for $k$-uniform hypergraphs, and let $\mathcal U_{\Delta,\geq k}$ be the analogous set for hypergraphs whose edge sizes are at least $k$. Thus $\mathcal U_\Delta=\mathcal U_{\Delta,2}$.

Zero-free locus conjecture. The zero-free locus of hypergraphs of maximum degree $\Delta$ is identical to the zero-free locus of graphs of maximum degree $\Delta$:

$$\mathcal U_{\Delta,\geq 2}=\mathcal U_\Delta.$$

This asks whether allowing arbitrary edge sizes gives no smaller zero-free locus than graphs. The paper notes that its general zero-free disk theorem gives a disk of radius $\lambda_s(\Delta+1)$ inside $\mathcal U_{\Delta,\geq 2}$, but the equality of the full loci remains open.