The extremal degree conjecture for eventually constant 0-Hecke operators

Let $S_n$ be the symmetric group, let $t_i:S_n\to S_n$ be the adjacent sorting operators, and let $H_0(S_n)=\langle t_1,\ldots,t_{n-1}\rangle$ be the $0$-Hecke monoid. An element $T\in H_0(S_n)$ is eventually constant if some positive power $T^k$ is the constant map with value $123\cdots n$. Let ${\bf B}:S_n\to S_n$ be bubble sort. Define $T_{\operatorname{odd}}$ and $T_{\operatorname{even}}$ by applying, respectively, the odd- and even-indexed operators in increasing order, and set $T_{\operatorname{tla}}=T_{\operatorname{odd}}\circ T_{\operatorname{even}}$.

Extremal degree conjecture. If $T\in H_0(S_n)$ is eventually constant, then

$$\deg({\bf B}:S_n\to S_n)\leq\deg(T:S_n\to S_n)\leq\deg(T_{\operatorname{tla}}:S_n\to S_n).$$

The conjecture asserts that bubble sort is closest to invertible and $T_{\operatorname{tla}}$ is farthest among eventually constant elements. The lower bound is explicit, since $\deg({\bf B}:S_n\to S_n)=n(n+1)/6$, while the paper says that the upper extremal degree is unknown; both inequalities remain open.