Presentation conjecture for Dehn quandles of Artin groups

Let $S$ be a generating set and let

$$\mathcal{A}=\langle S\mid (st)_{m_{st}}=(ts)_{m_{ts}}\text{ for }s,t\in S\text{ with }s\neq t\rangle$$

be an Artin group. For quandle elements, write $(s*t)_m$ for the alternating product of $s$ and $t$ under the quandle operation, beginning with $s$ and having $m$ factors. The presentation conjecture. The Dehn quandle $\mathcal{D}(S^{\mathcal{A}})$ has presentation

$$\mathcal{D}(S^{\mathcal{A}})=\langle S\mid (s*t)_{m_{st}}=s\text{ if }m_{st}\text{ is even, and }(t*s)_{m_{st}}=s\text{ if }m_{st}\text{ is odd for all }s\neq t\rangle.$$

The claim proposes a uniform presentation encompassing the spherical Artin-group and right-angled Artin-group examples discussed immediately beforehand. Its resolution would determine whether these parity-dependent relations always suffice to present the Dehn quandle of an Artin group.