Partition-function conjectures for quantum determinant and permanent singularities

Let $\mathcal{P}$ be the set of partitions, and define the partition functions

$$\vartheta_q^{\mathrm{det}}(t,\alpha)=\sum_{\lambda\in\mathcal{P}}\frac{m_q^\lambda(\alpha)}{f^\lambda}t^{|\lambda|},\qquad \vartheta_q^{\mathrm{per}}(t,\alpha)=\sum_{\lambda\in\mathcal{P}}\frac{m_q^\lambda(\alpha)_{\mathrm{per}}}{f^\lambda}t^{|\lambda|}.$$

Here $f^\lambda$ is the dimension of the symmetric-group module indexed by $\lambda$. A singular point is quantum when it is not principal. Partition-function conjecture. (1) If $\alpha$ is a quantum singular point and $k=1,2,\dots,\infty$, then

$$\vartheta_q^{\mathrm{det}}(t,\alpha)\ne\prod_{i=1}^{k}\frac{1}{1-t^i}.$$

(2) For $\alpha(q)\in\bigcup_{n=1}^{\infty}\operatorname{Sing}_{n,q}^{\mathrm{per}}$,

$$\vartheta_q^{\mathrm{det}}(t,\alpha(q^{-1}))=\vartheta_q^{\mathrm{per}}(t,\alpha(q)).$$

These are stated as slightly weaker reformulations of the determinant and reciprocity conjectures. The source does not provide a resolution of either assertion.