Maximal-rank conjecture for the cohomological map alpha_T

Let $V=\mathbb{C}^{n_1+1}\otimes\dots\otimes\mathbb{C}^{n_k+1}\otimes\mathbb{C}^{n+2}$, where $n=\sum_{j=1}^k n_j$, and let $T\in V$ be a general tensor. Let $\alpha_T$ be the cohomological map associated with the span $\langle Z_T\rangle\subset\mathbb{P}(H_T)$. Maximal-rank conjecture. The map $\alpha_T$ has maximal rank, namely

$$\operatorname{rank}(\alpha_T)=\min\left\{\prod_{j=1}^k\binom{n-1}{n_j},(n+2)\prod_{j=1}^k\binom{n-2}{n_j}\right\}.$$

The minimum is determined by dimensional considerations; this conjecture would imply the stated three-factor and higher-factor span conjectures. The paper reports full rank in all experiments except for the explicitly identified defective family, but does not establish the conjecture in general.