The limiting slope conjecture for degree covers of elliptic curves in genus three

Let $d$ be prime, let $\sigma$ be a partition of the branch-cycle type $(5^{1}1^{d-5})$, and let $Y_{3,d,\sigma}$ denote the corresponding one-parameter family of degree-$d$ covers of elliptic curves mapping to $\overline{\mathcal M}_{3}$. Its slope is denoted by $s(Y_{3,d,\sigma})$. The limiting slope conjecture. When $\sigma$ is of type $(5^{1}1^{d-5})$, one has

$$\lim_{d\rightarrow\infty}s(Y_{3,d,\sigma})=9.$$

Computer calculations for small prime values of $d$ suggest that the slope decreases to $9$, but the source provides no resolution of this asymptotic claim.