Depth-five classification conjecture for unramified multiple mixed values

Let $M(\bs;\varepsilon)$ denote a multiple mixed value and $T(\bs)$ a multiple t-value. Classify indices as wide, slender, or narrow according to whether their largest component is at least $4$, equal to $3$, or at most $2$. Depth-five MMV classification conjecture. In depth $5$, $S(2,1_3,4)$ is the only unramified wide MMV that is neither an MZV nor an MtV. Every unramified slender or narrow MMV must be an MZV, an MtV, belong to one of the two cited unramified families, or belong to the following list:

$$\begin{gathered} T(1,1,1,1,2),\ T(2,1,1,1,2),\ M(2,1,2,1,2;1_3,-1,1),\ M(1,2,1,2,2;1_2,-1,1_2),\\ T(1,2,1,1,2),\ T(1,1,2,1,2),\ M(1,2,2,1,2;1_3,-1,1),\ M(1,2,1,3,2;-1,1_4),\\ T(1,1,1,2,2),\ T(1,1,1,1,3),\ M(3,1,2,1,2;1_3,-1,1). \end{gathered}$$

The claim is motivated by the observation that $S(2,1_3,4)$ was found unramified in earlier work and is believed to be exceptional.