Digne–Gobet's conjecture on dual simple braids

Let $W$ be a finite Coxeter group with braid group $\mathbb{B}_W$, standard Garside monoid with Garside element $\Delta$, and dual positive monoid $\mathcal{B}e^+$. Write $[\Delta^{-1},\Delta]$ for the interval between $\Delta^{-1}$ and $\Delta$ in the standard Garside order. Digne–Gobet's conjecture. Every element $\beta\in\mathcal{B}e^+$ satisfies

$$\beta\in[\Delta^{-1},\Delta].$$

The conjecture concerns expressing simple elements of the dual Garside structure in terms of the classical Garside structure. Its relation to the standard and dual positive monoids is the motivating question described in the source; no resolution is given there.