Ultra log-concavity conjecture for dependence polynomials

Let $G$ be a graph, and let $D(G,x)$ denote its dependence polynomial.

Ultra log-concavity conjecture. For every graph $G$, $D(G,x)$ is ultra log-concave.

The paper proves ultra log-concavity for $K_2\cup 2K_1$-free graphs and graphs with an independent set of size $n-2$, but the assertion for arbitrary graphs remains open.