Clustering-basis conjecture for Jack superpolynomials

Let $\mathcal F_N^{(k)}$ be the ideal of symmetric superpolynomials over $\mathbb C$ in even variables $x_1,\ldots,x_N$ and odd variables $\ta_1,\ldots,\ta_N$ such that

$$f(x,\ta)=0\quad\text{whenever}\quad x_1=x_2=\cdots=x_{k+1}.$$

For admissible superpartitions, let $P_{\Lambda}^{(\alpha_{k,2})}(x_1,\ldots,x_N,\ta_1,\ldots,\ta_N)$ denote the Jack superpolynomials at parameter $\alpha_{k,2}$.

Clustering-basis conjecture. The Jack superpolynomials $P_{\Lambda}^{(\alpha_{k,2})}(x_1,\ldots,x_N,\ta_1,\ldots,\ta_N)$, with $\Lambda$ $(k,2,N)$-admissible, form a basis of $\mathcal F_N^{(k)}$. Equivalently, the inclusion $I_N^{(k,2)}\subseteq\mathcal F_N^{(k)}$ is an equality. The source gives no resolution status; the conjecture would identify the admissible Jack-superpolynomial realization of the clustering ideal.