Finite-generation conjecture for ${\cal B}_n^t$

Let ${\cal B}_n^t$ be the quotient of the braid algebra ${\cal B}_n$ with parameters $t_{ij}$, and let $\mathbb{Q}[t]=\mathbb{Q}[t_{ij},\ 1\leq i<j\leq n]$. Finite-generation conjecture. The algebra ${\cal B}_n^t$ is a finite-dimensional module over $\mathbb{Q}[t]$.

This is the braid-algebra analogue of the finite-generation claim for ${\cal E}_n^t$ and would imply strong control over the parameterized quotient. The source gives no resolution.