Non-closedness conjecture for symmetric tensors of bounded symmetric rank

Let $\mathcal{Y}_{r}$ be the set of symmetric tensors of symmetric rank at most $r$, and let $\overline{\mathcal{Y}}_{r}$ denote its closure. Let $R_{\mathsf{S}}$ be the maximum symmetric rank for order-$k$ symmetric tensors in $\mathbb{C}^{n}$. Non-closedness conjecture. Assume $k>2$ and $n\geq2$. Then

$$\mathcal{Y}_{r} \neq \overline{\mathcal{Y}}_{r}$$

for any $r$ such that $1<r<R_{\mathsf{S}}$. This strengthens the preceding propositions, which establish non-closedness when $R_{\mathsf{S}}<r$ or when $1<r\leq n$; the intermediate range $n<r<R_{\mathsf{S}}$ is conjectured to behave likewise.