Binary necklace-splitting conjecture for an arbitrary number of thieves

A necklace consists of beads of $n$ kinds, and $r$ thieves are placed at the vertices of a cube of dimension $\lceil\log_2 r\rceil$, allowing some vertices to remain unoccupied. A binary necklace splitting is a fair splitting in which adjacent, possibly degenerate, pieces are allocated to thieves whose vertices share an edge. The size of a splitting is its number of cuts.

Binary necklace-splitting conjecture. Given a necklace with $n$ kinds of beads and $r\geq 4$ thieves, there exists a binary necklace splitting of size $(r-1)n$.

The conjecture was previously posed for arbitrary $r$; the paper proves it when $r$ is a power of two, via the binary necklace-splitting theorem, so the general case remains open in the source.