The extremal-function classification conjecture for binary (3, t)-formations

Let $t\geq 6$. A $(3,t)$-formation is a concatenation of $t$ permutations of three distinct letters. Consider formations $u$ whose first permutation is $a b c$ and whose remaining permutations are each either $a b c$ or $a c b$. The formation-width $\operatorname{fw}(u)$ is the least integer $s$ such that every binary $(3,s)$-formation contains $u$ as a subsequence.

Extremal-function classification conjecture. Among these formations, $\operatorname{fw}(u)=2t-1$ if and only if $u$ is one of

$$(a b c)^{t},\quad a b c (a c b)^{t-1},\quad a b c (a c b)^{t-2} a b c,\quad a b c a c b (a b c)^{t-3} a c b,$$ $$a b c a c b (a b c)^{t-2},\quad a b c a b c (a c b)^{t-2},\quad (a b c)^{t-2} a c b a c b,\quad (a b c)^{t-2} a c b a b c,\quad (a b c)^{t-1} a c b.$$

This conjecture proposes a complete classification of the formations attaining the extremal value $2t-1$, extending the evidence and preceding proposition in the paper; its resolution would determine precisely which binary $(3,t)$-formations have maximal formation-width in this setting.