The four-strand simple pinning hardness conjecture

Let $\Sigma$ be a fixed orientable surface. The problem $\mathsf{SimplePin}(\Sigma,4)$ asks whether a simple multiloop with four strands on $\Sigma$ can be pinned with the specified input bound.

Hardness-starts-at-four conjecture. For a fixed orientable surface $\Sigma$, the problem $\mathsf{SimplePin}(\Sigma,4)$ is $\textsf{NP}$-complete.

Theorems in the paper establish polynomial-time solvability for at most three strands and $\textsf{NP}$-hardness for sufficiently many strands, motivating the conjectured threshold at four strands. The conjecture remains open.