Optimal alternating factorizations in groups with Lie algebra sl(2)

Let $n\in\mathbb{N}$ with $n\geq3$. Let $A_n(s)$ and $A(n;r,s,t)$ denote the alternating factorizations used in the paper, let $s_n$ be the associated switching parameters, and let $Z(G)$ be the center of the group $G$. Optimal alternating-factorization conjecture.

1. $A_n(s_n)$ is optimal in $\widetilde{\operatorname{SL}}(2,\mathbb{R})$ and in every group with $|Z(G)|\geq n-2$.

2. If $A(n;r,s,t)$ is optimal, then $s\geq s_{n-1}$; this holds for every group whose Lie algebra is $\mathop{\mathfrak{sl}}\nolimits(2)$.

3. If $|Z(G)|=N$, optimal alternating extremals have at most $N+3$ factors. If $A(N+3;r,s,t)$ is optimal, then

$$s\in[s_{N+2},s_{2N+2}].$$

These claims concern the optimality and factor-count restrictions for alternating extremals. The second claim is noted to be proved for $n=4$ and $n=5$, while the remaining assertions are presented as conjectural generalizations of the paper's arguments.