Entropy conjecture for positive pseudo-Anosov 4-braids

Let $\beta=\beta^+\Delta$ be a positive pseudo-Anosov 4-braid containing a half-twist and whose closure is a knot. Let $\lambda_{R(\beta)}$ be the 1-cocycle spectral radius, and let $\lambda_\beta$ denote the braid's associated quantity from the preceding construction. Entropy conjecture.

$$\lim_{m\to\infty}\lambda_{R(\beta^+\Delta^{2m+1})}=\lambda_\beta.$$

The conjecture relates the asymptotic spectral radius of the cocycle for increasingly twisted braids to the entropy-related invariant $\lambda_\beta$. The supplied text calls it optimistic and gives no evidence of resolution.