Conjecture on the ordering of weights in determinantal codes

Let $ell \leq m$ be positive integers and let $t$ be an integer satisfying $1<t<ell$. For the determinantal code, write $hat{w}_r(t;ell,m)$ for the weight indexed by $r$. Weight-ordering conjecture. The weights $hat{w}_1(t;ell,m),\dots,hat{w}_{ell}(t;ell,m)$ are mutually distinct; they satisfy

$$hat{w}_1(t;ell,m)<hat{w}_2(t;ell,m)<\dots<hat{w}_{ell-t+1}(t;ell,m),$$

and, for every $ell-t+2\leq rleqell$, the weight $hat{w}_r(t;ell,m)$ lies between $hat{w}_{r-2}(t;ell,m)$ and $hat{w}_{r-1}(t;ell,m)$. This conjecture concerns the unresolved comparison of nonzero weights when $1<t<ell$; the preceding special cases $t=1$ and $t=ell$ have simpler weight-ordering behavior.