The conjugation conjecture for fermionic-degree-one Macdonald superpolynomials

Let $\Lambda$ be a superpartition of fermionic degree $m=1$, and let $J_\Lambda$ be the integral form of the Macdonald superpolynomial. Let $S_\Omega$ be the deformed Schur superpolynomials, and define the linear map $\psi$ by

$$\psi(S_\Omega)=S_{\Omega^{\circledast}}.$$

Conjugation conjecture. The map $\psi$ satisfies

$$\psi(J_\Lambda)=J_{\Lambda^{\circledast}}.$$

This conjecture relates generalized Kostka coefficients under the superpartition conjugation operation in fermionic degree one. It is one of two proposed structural relations for these coefficients, with no resolution stated in the paper.