Closure-order conjecture for symplectic juggling-pattern cells

Let $JP(k,2n)^{sp}$ denote the set of symplectic $(k,2n)$-juggling patterns, let $X(k,2n)$ be the ambient variety, and let $X^{sp}(k,2n)$ be its symplectic locus. For $\mathcal{J},\mathcal{J}'in JP(k,2n)^{sp}$, write $C_{\mathcal{J}'}$ for the corresponding cell and $p_{\mathcal{J}}$ for the corresponding point.

Closure-order conjecture. If

$$p_{\mathcal{J}} \in \overline{C_{\mathcal{J}'}} \subset X(k,2n),$$

then

$$p_{\mathcal{J}} \in \overline{C_{\mathcal{J}'}^{sp}} \subset X^{sp}(k,2n).$$

This asserts that closure inclusion among symplectic cells is induced by closure inclusion among the corresponding cells in the ambient variety. Equivalently, the closure inclusion order on symplectic orbits should be the restriction of the closure inclusion order on the ambient orbits; the paper gives the claim as a conjecture after verifying it in the $(2,4)$ example.