Weyl-cycle conjecture for joins over secant varieties

Let $X^n_{n+3}$ be the blow-up of $\mathbb{P}^n$ at $n+3$ points. Let $L_I$ denote the linear space spanned by the relevant subset of base points, and let $\sigma_t$ denote the $t$-th secant variety to the rational normal curve of degree $n$ through the $n+3$ points. Define $J(L_I,\sigma_t)$ to be the corresponding join. Weyl-cycle conjecture. The varieties $J(L_I,\sigma_t)$ are Weyl cycles on $X^n_{n+3}$. This question extends the comparison between Weyl expected dimension and secant expected dimension. The paper notes that the claim agrees with the relevant description in dimension four, while its validity in arbitrary dimension is posed as a question.