S. Zhang's embedding conjecture for Hall algebra Drinfeld doubles

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories, and let $\widetilde{\mathbf{D}}\mathbf{H}_{\mathcal{C}_i}$ denote their reduced Drinfeld doubles. Let $\Psi:\mathbf{U}_{\nu}(\mathcal{L}\mathfrak{b}_+)\to\widetilde{\mathbf{H}}'_{\mathbb{X}_{\mathbf{p},\boldsymbol{\lambda}}}$ be the algebra morphism described in the preceding theorem. S. Zhang's embedding conjecture. The map $\Psi$ is an embedding for every weighted projective line $\mathbb{X}_{\mathbf{p},\boldsymbol{\lambda}}$. This extends the known embedding result for parabolic and elliptic weighted projective lines to the remaining cases.