The expected-dimension conjecture for higher-dimensional fradeco varieties

Let $\mathcal{T}_{r,n,d}\subset \mathbb{P}(\operatorname{Sym}_d(\mathbb{C}^n))$ be the fradeco variety, with $r>n$ and $d\geq 3$. Its expected dimension is

$$\min\left\{(n-1)(r-n)+\frac{(n-1)(n-2)}{2}+r-1,\,\binom{n+d-1}{d}-1\right\}.$$

Expected-dimension conjecture. The dimension of $\mathcal{T}_{r,n,d}$ equals

$$\min\left\{(n-1)(r-n)+\frac{(n-1)(n-2)}{2}+r-1,\,\binom{n+d-1}{d}-1\right\}$$

for all $r>n$ and $d\geq 3$. This would show that the natural parameterization has the expected image dimension; the source establishes only the corresponding upper bound.