Homogenized signature bound for positive braids

Let $B_b$ be the braid group on $b$ strands, let $\beta$ be a positive braid, and let $l(\beta)$ denote its positive word length. The asymptotic signature is the homogenization

$$\widetilde{\sigma}(\beta)=\lim_{i\to\infty}\frac{\sigma(\beta^i)}{i}.$$

Homogenized signature conjecture. For every positive braid $\beta$,

$$l(\beta)\geq-\widetilde{\sigma}(\beta)\geq\frac{1}{2}l(\beta).$$

This is presented as the homogenized analogue implied by the preceding conjecture. The supplied text does not state a resolution of this analogue, so its status remains open on the available evidence.