Bounded-dimensional support conjecture for minimal slice-rank decompositions

Let $d \ge 3$ and $k \ge 1$ be integers. An order-$d$ tensor $T$ over a field $\mathbb{F}$ has slice rank $k$ if it can be expressed using $k$ slices, where each slice has the form of a function in one variable multiplied by a function in the remaining $d-1$ variables. For each mode, write $[n_j]=\{1,\dots,n_j\}$, and let $\overline{x_j}$ denote the tuple of variables other than $x_j$.

Bounded-dimensional support conjecture. There exists a positive integer $C(d,k)$ such that, for every order-$d$ tensor $T$ over an arbitrary field $\mathbb{F}$ with slice rank $k$, there are linear subspaces

$$A_1 \subset \mathbb{F}^{n_1},\dots,A_d \subset \mathbb{F}^{n_d}$$

with dimension at most $C(d,k)$ such that the following holds: whenever $r_1,\dots,r_d$ are nonnegative integers satisfying

$$\nr_1+\dots+r_d=k,$$

and functions $a_{j,i}:[n_j]\to\mathbb{F}$, for $j\in[d]$ and $i\in[r_j]$, together with functions $b_{j,i}:\prod_{j'\ne j}[n_{j'}]\to\mathbb{F}$, satisfy

$$T(x_1,\dots,x_d)=\sum_{j=1}^d\sum_{i=1}^{r_j}a_{j,i}(x_j)b_{j,i}(\overline{x_j}),$$

then

$$\langle a_{1,1},\dots,a_{1,r_1}\rangle\subset A_1,\dots,\langle a_{d,1},\dots,a_{d,r_d}\rangle\subset A_d.$$

This would give a field-uniform bound on the subspaces supporting the one-variable factors in every minimal-length slice-rank decomposition. The question arises because the paper's preceding results do not provide such a uniform statement: over infinite fields the analogous theorem fails, while over finite fields the available bounds depend on the field and can grow square-exponentially in the slice rank. The source presents this as an open question rather than reporting a resolution.