Nonnegativity and log-concavity conjecture for the binomial coefficients of $\mathbb{Z}_2$ Hurwitz-Hodge integrals

Let $\vec{i}$ be a tuple of nonnegative integers, and let $c_{\vec{i},k}$ and $\widetilde{c}_{\vec{i},k}$ be the integers defined by

$$2^{|\vec{i}|+1}D_{(\vec{i}),2g+2}=\sum_{k=0}^{|\vec{i}|^2+1}c_{\vec{i},k}{g\choose k},\qquad 2^{|\vec{i}|+1}d_{(\vec{i}),2g+2}=\sum_{k=0}^{|\vec{i}|^2+1}\widetilde{c}_{\vec{i},k}{g\choose k}.$$

Nonnegativity and log-concavity conjecture. The integers $c_{\vec{i},k}$ and $\widetilde{c}_{\vec{i},k}$ are nonnegative, and, for fixed $\vec{i}$, the sequences $\{c_{\vec{i},k}\}_{k\geq0}$ and $\{\widetilde{c}_{\vec{i},k}\}_{k\geq0}$ are log-concave. The coefficients are known to exist and be integral by the stated binomial-basis result; the conjectured positivity and log-concavity are suggested by the computed examples.