Conjectural presentation of the extended double affine Artin group

Let $\ddot{X}_n$ be a double affine Coxeter diagram with finite nodes labelled $1,\ldots,n$. Let $B(X_n)$ be the Artin group of the finite subdiagram, and let $\mho$ and $\bar{\mho}$ be the finite groups of outer automorphisms associated to the affine nodes labelled by $\varTheta_0$ and $\varPhi_0$, respectively. Let $\omega$ and $\varpi$ denote their non-identity elements. Conjectural presentation. The group $\mathbf B(\ddot{X}_n)^e$ is generated by $B(X_n)$ and the finite groups $\mho$ and $\bar{\mho}$, subject to

$$\omega\mathcal T_i\omega=\mathcal T_{\omega(i)},\qquad \varpi\mathcal T_j\varpi=\mathcal T_{\varpi(j)}$$

for $1\leq i,j\leq n$ such that $\omega(i)$ and $\varpi(j)$ are finite nodes, and to the condition that

$$\mathcal D=\varpi\mathcal T_1\cdots\mathcal T_n\omega\varpi\mathcal T_n\cdots\mathcal T_1\omega$$

is central. The text presents this as the proposed presentation for the extended double affine Artin group of type $\ddot{X}_n$; no resolution status is supplied.