The quantitative Schmidt conjecture for dimensions of k-singular matrices

For integers $m,n$ with $(m,n)\neq(1,1)$, let $f_{m,n}(k)=mn-\frac{k(m+n-k)mn}{(m+n)^2}-\left\{\frac{km}{m+n}\right\}\left\{\frac{kn}{m+n}\right\}$, where $\{x\}$ denotes the fractional part of $x$. Let a matrix be $k$-singular when it satisfies the successive-minima condition referred to in the source. Quantitative Schmidt conjecture. For every $2\leq k\leq m+n-1$, the Hausdorff and packing dimensions of the set of $k$-singular $m\times n$ matrices are both

$$\max\big(f_{m,n}(k),f_{m,n}(k-1)\big).$$

The paper proves the corresponding lower bound and conjectures its optimality for both dimensions.