Equality of projected and ordinary SL_n-skein algebras

Establish (in full generality) that the projected ${\rm SL}_n$-skein algebra $_{\bar\omega}^{*}(S)$ equals the ${\rm SL}_n$-skein algebra $_{\bar\omega}(S)$ for punctured surfaces $S$ and arbitrary quantum parameter $\bar\omega$, beyond the cases already confirmed.

Background

The projected skein algebra is defined by quotienting the ordinary skein algebra by the kernel of the splitting homomorphism. Its equality with the original skein algebra was conjectured as a structural simplification with significant consequences for representation theory and Frobenius maps.

The paper notes that this conjecture has been proved for $n=2,3$ and for general $n$ under the condition $\bar\omega^n=1$, but does not claim a complete resolution in full generality. Completing the proof for all parameters would unify the framework and further simplify the use of Frobenius homomorphisms and quantum trace methods.