Fusion-lifting conjecture for formal ribbon extensions

Let $H$ be a finite-dimensional quasitriangular Hopf algebra and let $\tilde H$ be its formal ribbon extension. Let $M_1,M_2$ be indecomposable $H$-modules, and let $\tilde M_1,\tilde M_2$ be $\tilde H$-modules such that $\tilde M_1|_H\cong M_1$ and $\tilde M_2|_H\cong M_2$ as $H$-modules. Fusion-lifting conjecture. If the simple $H$-module $V$ is a composition factor of $M_1\otimes M_2$, then there is exactly one $\tilde H$-module $\tilde V$, up to isomorphism, such that $\tilde V|_H\cong V$ and $\tilde V$ is a composition factor of $\tilde M_1\otimes\tilde M_2$. If the indecomposable $H$-module $M$ is a direct summand of $M_1\otimes M_2$ as an $H$-module, then there is exactly one $\tilde H$-module $\tilde M$, up to isomorphism, such that $\tilde M|_H\cong M$ and $\tilde M$ is a direct summand of $\tilde M_1\otimes\tilde M_2$ as a $\tilde H$-module. The conjecture is supported by experimental evidence from the stated propositions and is also true when $H$ is ribbon; its general case remains open.