Zhu's unimodality conjecture for vertebrated graph independence polynomials

Let $V_n^{(m)}$ be the vertebrated graph, and let $I(V_n^{(m)};x)$ denote its independence polynomial. Zhu's conjecture. For $n,m$ with $n,m$ nonnegative integers, the polynomial $I(V_n^{(m)};x)$ is unimodal. The claim generalizes the stated results for centipedes and caterpillar graphs; the source does not give a resolution.