Optimality of short alternating factorizations in the universal cover of SL(2,R)

Let $A_n(s)$ be the alternating factorization defined in the paper, and let $\widetilde{\operatorname{SL}}(2,\mathbb{R})$ be the universal covering group of $\operatorname{SL}(2,\mathbb{R})$. Short-factorization optimality conjecture. The factorization $A_n(s)$ is optimal in $\widetilde{\operatorname{SL}}(2,\mathbb{R})$ for every $n\geq3$ and every

$$s\in[2,2\sqrt2].$$

This is a further optimality claim for alternating extremals, following the preceding discussion of central factorizations. The source gives no proof or resolution for the full stated range.