The orbit conjecture for apolar schemes to powers of quadrics

Let $q$ be the quadric and let $N(n,r)=\binom{n+r}{n}$ be the cactus rank of $q^r$. Denote by $\operatorname{VAPS}_G(q^r,N(n,r))$ the variety of smoothable apolar schemes of length $N(n,r)$ to $q^r$, let $\Gamma_0$ be the local apolar scheme, and let $SO(n,q)(\Gamma_0)$ be its orbit under the orthogonal group of $q$. Orbit conjecture.

$$\operatorname{VAPS}_G(q^r,N(n,r))=SO(n,q)(\Gamma_0),$$

when $n=2$, $r>3$, when $n=3,4$, $r>2$, and when $n>4$, $r>1$. The orbit of $\Gamma_0$ is already known to be contained in the variety, and the conjecture asserts that it exhausts the variety in the stated ranges.