Topological equivalence conjecture for 2,4 tangles

Let $l_1$ and $l_2$ be two $\rv{2,4}$ tangles, and let $L_1$ and $L_2$ be two arbitrarily chosen $\rv{2,4}$ tangle $n$-mosaic representatives of $l_1$ and $l_2$, respectively, with the same connection points along their boundaries. Topological equivalence conjecture for $\rv{2,4}$ tangles. The tangles $l_1$ and $l_2$ are ambient isotopic if and only if the representative mosaics $L_1$ and $L_2$ are $\rv{2,4}$ tangle mosaic equivalent.

This conjecture proposes that the defined boundary-adjusted mosaic equivalence exactly captures ambient isotopy of $\rv{2,4}$ tangles. The source provides no resolution, so the claim remains open.