Generalized Kruskal conjecture for product-vector sums

Let $n\geq 2$, $m\geq 1$, and $d_1,\dots,d_m\geq 1$ be integers. Let $\mathcal{X}_1,\dots,\mathcal{X}_m$ be vector spaces over a field $\mathbb{F}$, and let

$$\{x_a=x_{a,1}\otimes\dots\otimes x_{a,m}:a\in\{1,\dots,n\}\}\subset \mathcal{X}_1\otimes\dots\otimes\mathcal{X}_m$$

be non-zero product vectors such that, for each $j\in\{1,\dots,m\}$,

$$\dim\operatorname{span}\{x_{a,j}:a\in\{1,\dots,n\}\}\geq d_j.$$

If $n-1\leq\sum_{j=1}^m(d_j-1)$ and

$$\sum_{a=1}^n x_a=0,$$

then there exists a non-empty strict subset $\Gamma\subset\{1,\dots,n\}$ such that

$$\sum_{a\in\Gamma}x_a=0.$$

Generalized Kruskal conjecture. Under these hypotheses, a nontrivial zero-sum subcollection must exist. This relaxes general position to lower bounds on the dimensions of the subsystem spans and replaces pairwise cancellation by cancellation of an arbitrary non-empty strict subcollection. The stated consequence was previously proved, so this conjectural formulation is solved.