Parity-unimodality conjecture for fake-degree polynomials

For a partition $clambda$, let $f^{clambda}(q)$ be its fake-degree polynomial. A polynomial with coefficients $a_0,a_1,a_2,\cdots$ is parity-unimodal when the even-indexed coefficient sequence $a_0,a_2,a_4,\cdots$ and the odd-indexed coefficient sequence $a_1,a_3,a_5,\cdots$ are each unimodal.

Parity-unimodality conjecture. The fake-degree polynomials $f^{clambda}(q)$ are parity-unimodal for all partitions $clambda$.

The conjecture is motivated by results on $q$-Catalan polynomials and computational checks for $n\leq 50$. Its resolution is not indicated in the supplied text.