One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links.
This means that braids $B_1$ and $B_2$, considered inside a cube $I^3$, are related by ambient isotopy if and only if they are related by height-preserving ambient isotopy. This is a non-trivial fact whose proofs are all somewhat complicated (Stallings Theorem/ Magnus expansion/ embedding fibrations).
Given that there is no theoretical advantage to injecting braids into string links (distinct braids stay distinct), I wonder whether there is a computational advantage in doing so. Explicitly, given diagrams for $B_1$ and for $B_2$, is the minimum number of Reidemeister moves between them always realized for Reidemeister moves between braids? Or might the `shortest path between two braids' pass through string links?
Question: Is there an example of a pair of equivalent braid diagrams, considered as tangle diagrams, such that the minimum number of Reidemeister moves between them is increased if we allow only braid-like Reidemeister moves (i.e. if the result of each Reidemeister move must also be a braid)?