Polynomial codegrees and dual defects of skew-symmetric determinantal loci

Let $AD^{2s,2n}_{\binom{2n}{2}-m}$ denote the skew-symmetric determinantal locus in even dimension with the indicated rank parameter. For any fixed $m>0$, the even skew-symmetric determinantal-locus conjecture.

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

The conjecture is motivated by the known case $m=1$, whose codegree is polynomial in $n$ and whose dual defect is $\binom{2s}{2}-1$. The source also suggests an analogous statement in odd dimensions, but does not formulate it as a separate conjecture; no resolution is given.