Diagonal Supersymmetry conjecture for boson-fermion diagonal modules

Let $n$ be a positive integer, let $k$ and $j$ index the commuting and anticommuting variable sets, and let $\mathcal{E}_n(\bm{q};\bm{z})$ be the generic Frobenius characteristic of the pure bosonic multivariate coinvariant module, with

$$\mathcal{E}_n(\bm{q};\bm{z})=\sum_{\mu\vdash n}\sum_\lambda c_{\lambda\mu}s_\lambda(\bm{q})s_\mu(\bm{z}).$$

Write $\varepsilon$ for the plethystic sign satisfying $p_i[\varepsilon]=(-1)^i$, and let $\mathbb{DBF}_{k,j;n}(\bm{q};\bm{u};\bm{z})$ denote the multigraded Frobenius characteristic of the boson-fermion diagonal module. Diagonal Supersymmetry conjecture. The characteristic is obtained universally from the bosonic one by

$$\mathbb{DBF}_{k,j;n}(\bm{q};\bm{u};\bm{z})=\mathcal{E}_n(\bm{q}-\varepsilon\,\bm{u};\bm{z})$$

and hence

$$\mathbb{DBF}_{k,j;n}(\bm{q};\bm{u};\bm{z})=\sum_{\mu\vdash n}\sum_\lambda c_{\lambda\mu}s_\lambda[\bm{q}-\varepsilon\,\bm{u}]s_\mu(\bm{z}).$$

This is presented as the paper's main conjecture, asserting that all boson-fermion cases arise from the generic bosonic characteristic by plethystic substitution; the supplied text gives no resolution status.