Higher-factor defective-span conjecture beyond boundary format

Let $n=\sum_{j=1}^k n_j$ and consider a $(k+1)$-tensor space $V$ of format $(n_1+1,\dots,n_k+1,n+2)$, with $T\in V$ a general tensor. Higher-factor defective-span conjecture. If $k\geq3$, then

$$\langle Z_T\rangle=\mathbb{P}(H_T)$$

except in the cases $(2,2,2,5)$, $(2,2,3,6)$, and $(2,2,4,7)$. In these latter cases,

$$\langle Z_T\rangle\subsetneq\mathbb{P}(H_T)$$

has codimension $\prod_{j=1}^k\binom{n-1}{n_j}-(n+2)\prod_{j=1}^k\binom{n-2}{n_j}$. This conjecture is the higher-factor consequence of the maximal-rank conjecture for $\alpha_T$ and identifies the listed formats as the only predicted defective cases.