Topological equivalence conjecture for tangles

Let $t_1$ and $t_2$ be tame tangles, and let $T_1$ and $T_2$ be arbitrarily chosen mosaic representatives of $t_1$ and $t_2$, respectively. Topological equivalence conjecture for tangles. The tangles $t_1$ and $t_2$ are of the same tangle type if and only if, after applying finitely many tangle injections to place the representatives in a common dimension if necessary, the resulting mosaics are related by a finite sequence of mosaic Reidemeister moves.

This conjecture asserts injectivity of the realization map from mosaic tangle equivalence classes to topological tangle equivalence classes, complementing the stated surjectivity for tame tangles. Its resolution is not supplied in the source.