Conjectural presentation of the extended triple affine Artin group

Let $\dddot{X}_n$ be a double affine Coxeter diagram, let $B(X_n)$ be the Artin group associated to its finite subdiagram, and let $\mho_{01}$, $\mho_{02}$, and $\mho_{03}$ be the finite groups of outer automorphisms associated to the three affine nodes. Let $I$ be the set of finite labels of the images of the affine nodes, let $i^*$ satisfy $\omega_{01,i^*}=\omega_{01,i}^{-1}$, and write $\mathcal T_k$ and $\mathcal T_{w_{i^*}}$ for the corresponding elements of $B(X_n)$. Conjectural presentation. The group $\mathbf B(\dddot{X}_n)^e$ is generated by $B(X_n)$ and the finite groups $\mho_{01}$, $\mho_{02}$, and $\mho_{03}$, subject to

$$\omega_{0j,i}\mathcal T_k\omega_{0j,i}^{-1}=\mathcal T_{\omega_{0j,i}(k)}$$

for $1\leq j\leq 3$, $i\in I$, $1\leq k\leq n$, and $k\neq i^*$, and to the condition that

$$\mathcal D_i=\omega_{01,i}\omega_{02,i^*}\omega_{03,i}\mathcal T_{i^*}\mathcal T_{w_{i^*}}$$

is central for every $i\in I$. This is an alternative conjectural presentation of the extended double affine Artin group for diagrams of type $\dddot{X}_n$; the supplied text does not state whether the presentation has been proved or disproved.