Generator conjecture for the $\mathfrak{sl}_2$ bimodules $\mathcal{H}_{\lambda,\mu}$

Let $H$ be the Hopf algebra used in the $\mathfrak{sl}_2$ example, with generators $K,b,c$. For integers $\lambda$ and $\mu$, let $\mathcal{H}_{\lambda,\mu}$ be the corresponding highest-weight bivector space and define

$$h_{\lambda,\mu}=\begin{cases}K^{-(\lambda+\mu)/2}c^{\lambda-\mu}&\text{if }\lambda\geq\mu,\K^{-(\lambda+\mu)/2}b^{\mu-\lambda}&\text{if }\lambda<\mu.\end{cases}$$

Bimodule generation conjecture. The $D(H)$-bimodule $\mathcal{H}_{\lambda,\mu}$ is generated by $h_{\lambda,\mu}$. The conjecture is based on the examples computed in the paper, and the source gives no resolution.