D., Fromentin and Gebhardt's formula for the function $\mu_3$

Let $B_3^+$ be the positive braid monoid on three strands, let $\Delta_3$ denote its fundamental braid, and let $\mu_3$ be the function on $B_3^+$ introduced in the paper. D., Fromentin and Gebhardt's conjecture. For every $\beta\in B_3^+$, one has

$$\mu_3(\beta\Delta_3^2)=\sigma_1\sigma_2^2\sigma_1\cdot\mu_3(\beta)\cdot\sigma_1^2.$$

The formula is suggested by computer experiments and an analogous formula for $n$ strands is described as easy to guess; the supplied text gives no resolution status.