Arrondo–Bernardi nondefectivity conjecture for secant varieties of Chow varieties

Let $n,d,s$ be positive integers, and let $\operatorname{Split}_d(\mathbb P^n)$ denote the Chow variety of degree-$d$ forms in $n+1$ variables that split completely into products of linear forms. Its $s$th secant variety is denoted by $\sigma_s(\operatorname{Split}_d(\mathbb P^n))$. Define its expected dimension by

$$\operatorname{expdim}\sigma_s(\operatorname{Split}_d(\mathbb P^n))=\min\left\{s(dn+1),\binom{n+d}{d}\right\}-1.$$

The secant variety is nondefective when

$$\dim\sigma_s(\operatorname{Split}_d(\mathbb P^n))=\operatorname{expdim}\sigma_s(\operatorname{Split}_d(\mathbb P^n)).$$

Arrondo–Bernardi conjecture. The secant variety $\sigma_s(\operatorname{Split}_d(\mathbb P^n))$ is nondefective unless $d=2$ and $2\leq s\leq\frac{n}{2}$.

This conjecture predicts that the expected dimension obtained by a naive parameter count holds for secant varieties of Chow varieties, apart from the stated quadratic exceptions. It is presented as a conjecture of Arrondo and Bernardi; the supplied text gives no resolution status.