Conjecture on secant varieties of Segre products with completely decomposable forms

Let $\operatorname{Split}_d(\mathbb P^n)$ be the variety of degree-$d$ completely decomposable forms, let $\operatorname{Seg}(\mathbb P^m\times\operatorname{Split}_d(\mathbb P^n))$ be their Segre product, and let $\sigma_s$ denote its $s$-th secant variety. The secant variety is nondefective when it has the expected dimension

$$\min\left\{s(m+dn+1),(m+1)\binom{n+d}{d}\right\}-1.$$

Segre–split secant conjecture. If $m\geq 1$, then $\sigma_s(\operatorname{Seg}(\mathbb P^m\times\operatorname{Split}_d(\mathbb P^n))$ is nondefective for all cases except those outlined in the known unbalanced family of Proposition }. The conjecture is based on Macaulay2 computations and would identify all defective cases in this family beyond the stated unbalanced exception.