Projected stated (SL_n)-skein algebras and the Huang–Wang conjectures

We prove Conjectures 1.5 and 1.6 of Min Huang and Zhihao Wang: the projected stated (SL_n)-skein algebra equals its quantum cluster and upper cluster algebras, and the specified exchangeable variables generate it over the frozen Laurent coefficient algebra. The results hold for triangulable unpunctured marked surfaces with at least two boundary marked points per connected component, over every coefficient domain and parameter in the conjectures.