Continuity and computability conjecture for dimensions of nontrivially singular matrices

For $m,n$ positive integers, let $\operatorname{Sing}_{m,n}^*(\omega)$ denote the set of $m\times n$ matrices with Diophantine exponent at least $\omega$ that are not trivially singular, and let $\operatorname{HD}$ and $\operatorname{PD}$ denote Hausdorff and packing dimension, respectively. Continuity and computability conjecture. The functions

$$\omega\mapsto\operatorname{HD}(\operatorname{Sing}_{m,n}^*(\omega)), \qquad \omega\mapsto\operatorname{PD}(\operatorname{Sing}_{m,n}^*(\omega))$$

are decreasing and continuous, and are computable in the sense of Weihrauch. This would give regularity and effective computability for the dimension functions arising from the paper's variational principle. The supplied text presents the assertion in a conjecture environment but gives no evidence of a resolution.