The braid-to-tangle skein isomorphism conjecture for the punctured torus

Let $\mathrm{BSk}_n(T^2,*)$ be the braid skein algebra of the punctured torus with $n$ strands, and let $\mathrm{Sk}_n(T^2,*)$ be the corresponding full framed Homflypt tangle skein algebra. There is a natural homomorphism from the braid skein algebra to the tangle skein algebra, obtained by viewing braids with consistent framings as framed tangles. Braid-to-tangle skein isomorphism conjecture. The map

$$\mathrm{BSk}_n(T^2,*)\to\mathrm{Sk}_n(T^2,*)$$

from the braid skein algebra to the tangle skein algebra is an isomorphism. This would show that allowing general framed arcs introduces no additional algebraic relations or collapse beyond the braid skein description.