The incomparability conjecture for multigraded Frobenius series of coinvariant rings

For fixed $n$ and nonnegative integers $k,j$, let $R_n^{(k,j)}$ be the corresponding multigraded coinvariant ring, and write

$$\operatorname{Frob}(R_n^{(k,j)}; \mathbf{q};\mathbf{u}) = \sum_{\lambda \in P(k,j,n)}\sum_{\mu \vdash n} c_{\lambda\mu} s_\lambda(\mathbf{q}/\mathbf{u})s_\mu(\mathbf{z}).$$

For $0\leq k'\leq k$ and $0\leq j'\leq j$, restriction determines the coefficients needed for $R_n^{(k',j')}$. Two pairs are incomparable in the componentwise order when neither is componentwise at most the other. The incomparability conjecture. When $(k,j)$ and $(k',j')$ are incomparable in the componentwise order, the expansion of the multigraded Frobenius series of $R_n^{(k',j')}$ contains nonzero coefficients $c_{\lambda\mu}$ that are not determined by the expansion of the multigraded Frobenius series of $R_n^{(k,j)}$. This conjecture asserts that incomparable parameter pairs contain genuinely independent Frobenius data; whether this always occurs remains open.