Affine orbit-closure conjecture for reparametrisation groups

Let $p\geq 1$, let $k$ be sufficiently large depending on $p$, let $n=\mathrm{sym}^{\leq k}p$, and let $M\in\mathbb{N}$. Set

$$K=M\left(\sum_{i=1}^k i\,\mathrm{sym}^i p\right)+1.$$

With $p_{k,p}$ as defined by

$$p_{k,p}=\ldots \wedge \sum_{|\mathbf{s}|=j}\sum_{\mathbf{s}_1+\mathbf{s}_2+\ldots+\mathbf{s}_j=\mathbf{s}}e_{\mathbf{s}_1}e_{\mathbf{s}_2}\ldots e_{\mathbf{s}_j}\wedge\ldots,$$

consider the point

$$p_{k,p}\otimes(e_1\wedge\ldots\wedge e_p)^{\otimes K}\in\wedge^n(\,\operatorname{Sym}^{\leq k}\mathbb{C}^n)\otimes(\wedge^p(\mathbb{C}^n))^{\otimes K}.$$

Affine orbit-closure conjecture. This point has stabiliser $\mathbb{G}_{k,p}'$ in $\mathrm{SL}(n)$, and the closure of its $\mathrm{SL}(n)$-orbit is the union of its orbit and finitely many other $\mathrm{SL}(n)$-orbits, all of codimension at least two if $k$ is large enough depending on $p$ and $M$ is sufficiently large depending on $k$ and $p$.

The conjecture would give an affine embedding of $\mathrm{SL}(n)/\mathbb{G}_{k,p}'$ whose boundary has codimension at least two, generalising the preceding theorem and reducing the invariant-theoretic problem to the geometry of this orbit closure.