Snake in the Snail conjecture for minimal snake modules

Let $n$ be the rank parameter, let $\mu\in\mathbb{C}$ be fixed, and let $S^{(k)}_1(\mu-(k-1)(n+1)/2)$ denote the minimal snake module appearing in the snail operator $\tilde{X}_k$. The extended T-systems are the relations

$$[S^{(1)}_{m+1}(\mu-l\tfrac{n+1}{2})][S^{(l)}_m(\mu-(l-1)\tfrac{n+1}{2})]=[S^{(l+1)}_{m+1}(\mu-l\tfrac{n+1}{2})]+[S^{(l-1)}_{m+1}(\mu-(l-2)\tfrac{n+1}{2})].$$

Snake in the Snail conjecture. For odd $m$, the extended T-systems appear in successive lines of the snail operator, and the component corresponding to

$$[S^{(l-1)}_{m+1}(\mu-(l+2)\tfrac{n+1}{2})]$$

cancels. Consequently, the snail operator $\tilde{X}_k$ can be defined through a single irreducible representation of the Yangian, as in the $\mathfrak{sl}_2$ case, namely the minimal snake module $S^{(k)}_1(\mu-(k-1)(n+1)/2)$. This conjecture describes the hoped-for simplification of the snail operator from a composite construction to one irreducible Yangian representation; the supplied text gives no resolution, so its status remains open.