Gallotti–Pierre-Louis conjecture on circular elastic knots

Let $N$ be a positive integer, and let a knot class have braid index and bridge index both equal to $N$. An $N$-fold circle is the corresponding circle traversed $N$ times, and an elastic knot is understood in the sense of Gerlach–Reiter–von der Mosel. Circular elastic knots conjecture. The $N$-fold circle is the unique elastic knot for any knot class whose braid index and bridge index coincide with $N$. This is an important open problem in elastic knot theory; the conjecture was first stated by Gallotti and Pierre-Louis and later formulated mathematically by Gerlach–Reiter–von der Mosel.