Berenstein–Kazhdan's presentation conjecture for the algebra ${\bf D}(W)$

Let $W$ be a crystallographic Coxeter group, let ${\mathcal S}$ be its set of reflections, and let ${\bf D}(W)$ be the image of the algebra generated by the elements $D_s$ for $s\in{\mathcal S}$. For distinct $i,j\in I$, write $m=m_{ij}\ge2$, let $n$ be a divisor of $m$, let $1\le r\le n$, and define

$$D_k=D_{w\cdot s_is_j\cdots s_i\cdot w^{-1}}\in{\bf D}(W),$$

where the alternating word has length $2k-1$, $k=1,\ldots,m$, and $w\in W$ satisfies $\ell(ws_i)=\ell(w)+1$ and $\ell(ws_j)=\ell(w)+1$.

Berenstein–Kazhdan's presentation conjecture. The algebra ${\bf D}(W)$ is generated by the $D_s$ subject to the idempotent relations $D_s^2=D_s$, the braid relations

$$D_iD_jD_i\cdots=D_jD_iD_j\cdots$$

with $m_{ij}$ factors on each side, and the quadratic-linear and Yang–Baxter-type relations stated in the source:

$$\sum_{0\le a<b<m/n\,:\,b-a=m/n-p}D_{r+an}D_{r+bn}=\sum_{0\le a'<b'<m/n\,:\,b'-a'=p}D_{r+b'n}D_{r+a'n}-\sum_{p\le c<m/n-p}D_{r+cn}$$

for $1\le p<m/(2n)$, together with

$$\overrightarrow{\prod_{t\le a\le m/n-1}}(1-D_{r+an})\,\overrightarrow{\prod_{0\le b\le t-1}}D_{r+bn}=\overleftarrow{\prod_{0\le b\le t-1}}D_{r+bn}\,\overleftarrow{\prod_{t\le a\le m/n-1}}(1-D_{r+an})$$

for $0\le t\le m/n$.

The source attributes this presentation conjecture to Berenstein and Kazhdan. The supplied parser gives no resolution evidence, and the paper's surrounding text moves on to another result without explicitly stating that this conjecture is proved, so its database status remains open.