Strassen's rank-preserving pair conjecture for Segre varieties

Let $A_j=A_j'\oplus A_j”$ be decompositions of vector spaces, and set

$$L=\mathbb P\bigl((A_1'\otimes\cdots\otimes A_k')\oplus(A_1”\otimes\cdots\otimes A_k”)\bigr).$$

Let $X=\operatorname{Seg}(\mathbb P A_1\times\cdots\times\mathbb P A_k)$, and let $R_X$ denote the $X$-rank. Strassen's rank-preserving pair conjecture. The pair $(X,L)$ is rank preserving: if $Y=(X\cap L)_{\operatorname{red}}$, then $\langle Y\rangle=L$ and $R_X|_L=R_Y$. The supplied text presents this as a question/conjecture of Strassen and does not give a resolution.