Rigidity conjecture for braided symmetric and exterior cubes

Let $\mathfrak{g}$ be a simple Lie algebra, let $V^q$ be a simple $U_q(\mathfrak{g})$-module, and let $V$ be its classical limit. The degree-three braided symmetric and exterior powers are $S^3_\sigma V^q$ and $\Lambda^3_\sigma V^q$, while their classical low-degree counterparts are $S^3_{low}V$ and $\Lambda^3_{low}V$. Rigidity conjecture. The classical limit of $S^3_\sigma V^q$ (respectively, $\Lambda^3_\sigma V^q$) is isomorphic to $S^3_{low}V$ (respectively, $\Lambda^3_{low}V$) as a $U(\mathfrak{g})$-module. The conjecture concerns rigidity of braided symmetric and exterior cubes under passage to the classical limit. The supplied status evidence says the assertion is disproved in the nonsimple setting, while the stated conjecture has simple $V^q$ and simple $\mathfrak{g}$; its status for precisely these hypotheses is not resolved by the supplied evidence.