Higher braided symmetric powers conjecture for $U_q(gl_2(\mathbb{C}))$-modules

Let $V_\ell$ denote the $U_q(gl_2(\mathbb{C}))$-module indexed by $\ell\geq 0$, and let $S_\sigma^n V_\ell$ be its $n$th braided symmetric power. Higher braided symmetric powers conjecture. For any $\ell\geq 0$ and any $n\geq 4$, one has

$$S_\sigma^n V_\ell \cong \begin{cases} \displaystyle\bigoplus_{0\leq i\leq \frac{\ell-1}{2}} V_{(n\ell-2i,2i)} & \text{if $\ell$ is odd},\\ \displaystyle\bigoplus_{0\leq i\leq \frac{n\ell}{4}} V_{(n\ell-2i,2i)} & \text{if $\ell$ is even}. \end{cases}$$

In particular, $S_\sigma^n V_\ell$ is multiplicity-free and, for $n\geq 4$,

$$\dim S_\sigma^n V_\ell= \begin{cases} \displaystyle\frac{(\ell+1)(\ell(n-1)+2)}{2} & \text{if $\ell$ is odd},\\ \displaystyle\binom{\frac{n\ell}{2}+2}{2} & \text{if $\ell$ is even}. \end{cases}$$

The conjecture extends the paper's explicit description of the braided symmetric cube to all higher powers; the cases $n=3$ are established in the preceding theorem, while the source gives no resolution for $n\geq 4$.