Secant-dimension conjecture for varieties of reducible forms

Let $\Bbbk$ be an algebraically closed field of characteristic zero, let $n\geq 1$, let $\mathcal{S}=\bigoplus_{d\geq 0}\mathcal{S}_d=\Bbbk[x_0,\ldots,x_n]$, and fix $d\geq 2$. For $1\leq j\leq d/2$, let $X_j\subseteq\mathbb{P}\mathcal{S}_d$ be the variety of degree-$d$ forms with a degree-$j$ factor, and let $X_{\operatorname{red}}=\bigcup_{j=1}^{\lfloor d/2\rfloor}X_j$ be the variety of reducible forms. For $r\geq 1$, let $\sigma_r(X)$ denote the $r$th secant variety of $X$.

Secant-dimension conjecture. For each integer $r\geq 1$, we have

$$\dim \sigma_r(X_{\operatorname{red}})=\dim \sigma_r(X_1).$$

This is presented as a stronger conjecture implying equality of the generic strength and slice rank. The source attributes it to the cited work and states that proving it is the aim of the paper; consequently it is solved in the source context.