Nondefectivity conjecture for secant varieties of Chow–Veronese varieties

Let $\boldsymbol d$ be a composition of the degree data, let $n$ be a nonnegative integer, and let $\operatorname{CV}_{\boldsymbol d}(\mathbb P^n)$ denote the corresponding Chow–Veronese variety. For its $s$th secant variety, write

\operatorname{expdim}\sigma_s(\operatorname{CV}_{\boldsymbol d}(\mathbb P^n))=\min\left\{s(kn+1),\genfrac{0pt}{}{n+d}{d}\right\}-1,

where $k$ is the number of parts of $\boldsymbol d$ and $d=\lVert\boldsymbol d\rVert_1$. Nondefectivity conjecture. With the exception of the known defective cases, one has

$$\dim\sigma_s(\operatorname{CV}_{\boldsymbol d}(\mathbb P^n))=\operatorname{expdim}\sigma_s(\operatorname{CV}_{\boldsymbol d}(\mathbb P^n))$$

for all $\boldsymbol d$, $n$, and $s$. Thus, apart from the listed exceptions, the secant varieties have the dimension predicted by the naive parameter count. The conjecture concerns the classification of defective secant varieties of Chow–Veronese varieties; the supplied text identifies known defective cases but gives no resolution of the assertion in full generality.