The Galois-width conjecture for determinantal maximum-likelihood covers

Let $X_{m,n,r}$ and $Y_{m,n,r}$ denote the corresponding rectangular determinantal varieties and their dual varieties, and let $Z_{n,r}$ be the symmetric determinantal variety defined by the rank condition in the preceding setup. Write $\operatorname{gw}(\pi)$ for the Galois width of a branched cover $\pi$, and $\operatorname{ML}(V)$ for the maximum-likelihood degree of a variety $V$. For integers $1\le r\le n\le m$, the conjecture asserts

Galois-width conjecture.

$$\operatorname{gw}\left(\pi_{\mathcal{L}_{X_{m,n,r}}}\right)= \begin{cases} \operatorname{ML}(Y_{m,n,r})/2 & \text{if } n \text{ is odd and } r=(n+1)/2,\\ \operatorname{ML}(Y_{m,n,r}) & \text{otherwise,} \end{cases}$$

and

$$\operatorname{gw}\left(\pi_{\mathcal{L}_{Z_{n,r}}}\right)= \begin{cases} \operatorname{ML}(Z_{n,r})/2 & \text{if } n \text{ is odd and } r=(n+1)/2,\\ \operatorname{ML}(Z_{n,r}) & \text{otherwise.} \end{cases}$$

The conjecture predicts that the known order-two duality accounts for exactly the only systematic reduction of Galois width from maximum-likelihood degree in the stated families. The supplied text does not establish the general formulas or give evidence resolving them.