Quantum-group compatibility of the abelianization map

For a fibered knot $K$, let

$$\mathrm{Ab}:\operatorname{Sk}^{\mathrm{GL}_2}_q(S^3\setminus K)\longrightarrow\operatorname{Sk}^{\mathrm{GL}_1}_q(S^3\setminus K)^{\otimes2}$$

be the abelianization homomorphism from the theorem in the paper.

Quantum-group abelianization conjecture. The abelianization map $\mathrm{Ab}$ is equivalent to the abelianization map constructed using quantum groups.

This conjecture identifies the geometric abelianization map with the quantum-group construction. The comparison is formulated using the corresponding $\mathrm{SL}_2$-skein construction after the stated framing normalization.