Polynomial codegrees and dual defects of general determinantal loci

Let $D^{s,n}_{n^2-m}$ denote the determinantal locus of general $n\times n$ matrices of the indicated rank parameter. For any fixed $m>0$, the general determinantal-locus conjecture.

$$\begin{aligned} &\text{the dual defect of }D^{s,n}_{n^2-m}\text{ is }\max(0,s^2-m),\text{ independently of }n,\\ &\text{and the codegree of }D^{s,n}_{n^2-m}\text{ depends polynomially on }n. \end{aligned}$$

The conjecture extends the known case $m=1$, where the dual defect is $s^2-1$ and the codegree is given by a polynomial expression in $n$. It predicts uniform behavior for defective general determinantal loci with fixed $m$; the source gives no resolution.