Dimension formula for symmetric spaces times regular trees

Let $X$ be a symmetric space with trivial compact factor, and let $\ell\in\mathbb{N}$. Here $T_3$ denotes the regular tree of valence $3$, and $\operatorname{corank}(X)$ is the corank invariant. Dimension formula conjecture.

$$\operatorname{asdim}_{\operatorname{poly}}(X\times T_3^\ell)=\operatorname{asdim}_{\operatorname{se}}(X\times T_3^\ell)=\operatorname{corank}(X)+\ell.$$

The conjecture concerns the relative asymptotic dimensions introduced in the paper and would identify both of them for products of symmetric spaces with regular trees. Its status is not specified in the source.