Vanishing and log-concavity conjecture for $\mathbb{Z}_2$ Hurwitz-Hodge integrals

Let $g>0$ and let $\vec{\ell}\in\mathbb{Z}_{\geq0}^n$ satisfy $|\vec{\ell}|\leq g-1$, where $|\vec{\ell}|$ is the sum of the entries of $\vec{\ell}$. Let $D_{(\vec{\ell},i),2g+2}$ and $d_{(\vec{\ell},i),2g+2}$ denote the corresponding Hurwitz-Hodge integrals. Vanishing and log-concavity conjecture. One has

$$\sum_{i=0}^g(-2)^iD_{(\vec{\ell},i),2g+2}=0,$$

and the sequences $\{2^{|\vec{\ell}|+i+1}D_{(\vec{\ell},i),2g+2}\}_{i\geq0}$ and $\{2^{|\vec{\ell}|+i+1}d_{(\vec{\ell},i),2g+2}\}_{i\geq0}$ are log-concave. A special case of the vanishing assertion is known from earlier work, and the source notes that the general vanishing follows from results of Afandi; the proposed extension combines this vanishing with the conjectured log-concavity.