Sturmfels's determinantal ideal conjecture for tensor train varieties

Let $\mathbf{k}=(k_0,\dots,k_n)$ and $\mathbf{r}=(r_1,\dots,r_n)$, let $\psi$ be a $k_0\times\cdots\times k_n$ tensor of indeterminates, and let $\psi^i$ denote its flattenings. The tensor train variety $V_{\mathbf{k},\mathbf{r}}$ is the variety parametrized by tensor trains with format $\mathbf{k}$ and rank bounds $\mathbf{r}$. Sturmfels's conjecture. The homogeneous prime ideal of $V_{\mathbf{k},\mathbf{r}}$ in $\mathbb{C}[\psi]$ is

$$\mathcal I(V_{\mathbf{k}, \mathbf{r}}) = \sum_{i=0}^{n-1} \Big\langle (r_{i+1}+1)\textrm{-minors of }\psi^i \Big\rangle.$$

Moreover, the union of the $(r_{i+1}+1)$-minors of the flattenings $\psi^i$ for $i=0,\dots,n-1$ forms a Gröbner basis for $\mathcal I(V_{\mathbf{k},\mathbf{r}})$. This would upgrade the known set-theoretic determinantal description to a scheme-theoretic one; it was suggested by Bernd Sturmfels, and remains open in general.