Landesman's homological stability conjecture for Selmer spaces

For fixed $n$, let $\operatorname{Sel}_n^d(\mathbb{C})$ be the Selmer space associated with the universal family of Weierstrass models of degree parameter $d$. A Landesman homological stability conjecture. There are constants $A$ and $B$ depending on $n$ such that

$$\dim H_i(\operatorname{Sel}_n^d(\mathbb{C});\mathbb{Q})=\dim H_i(\operatorname{Sel}_n^{d+1}(\mathbb{C});\mathbb{Q})$$

whenever $d\geq Ai+B$. This prediction is framed as homological stability for Selmer spaces as $d$ tends to infinity and would likely imply the associated number-theoretic conjecture concerning average sizes of Selmer groups of elliptic curves over function fields; the source does not state whether it has been proved.