The non-orientable surface pinning complexity conjecture

Let $\Sigma$ be a non-orientable surface and let $s\in\mathbb N$. The problems $\mathsf{Pin}(\Sigma,s)$ and $\mathsf{SimplePin}(\Sigma,s)$ are the pinning and simple-pinning decision problems for multiloops with $s$ strands on $\Sigma$.

Non-orientable surface complexity conjecture. Both $\mathsf{Pin}(\Sigma,s)$ and $\mathsf{SimplePin}(\Sigma,s)$ are in $\textsf{P}$ when $s\leq 3$, and are $\textsf{NP}$-complete when $s\geq 4$.

The paper's results rely on theorems for orientable surfaces, so the corresponding complexity classification for non-orientable surfaces is left as a further direction.