Unramified MTV classification by width and dual depth

Let $\bfs$ be an index, let $T(\bfs)$ be a multiple t-value (MTV), and let $\bfs^*$ be its dual. Define the width $\operatorname{wth}(\bfs)$ as the largest component of $\bfs$; call $\bfs$ wide, slender, or narrow according as its width is at least $4$, equal to $3$, or at most $2$, respectively. Width classification conjecture. A wide MTV of depth at least $4$ is ramified. A slender MTV is unramified if and only if it has one of the forms $T(1_{2k},3,1_{2l},2)$ or $T(1_{2k},3)$ for some $k,l\in\mathbb{N}\cup\{0\}$. A narrow MTV is unramified if and only if it has one of the forms $T(1_k,2)$ or $T(2,1_{2k+1},2)$ for some $k\in\mathbb{N}\cup\{0\}$; $T(2,1_k,2,1_l)$ for some $k,l\in\mathbb{N}\cup\{0\}$ with $k+l$ odd; or $T(1_{2k+1},2,1_{2l},2,1_{2m+1})$ for some $k,l,m\in\mathbb{N}\cup\{0\}$. This conjecture would imply that an MTV and its dual cannot both have depth at least $4$ while remaining unramified; the source notes that this was previously conjectured by Kaneko and Tsumura.