The unpinning avoidance game PSPACE-completeness conjecture

Let $\gamma\colon\bigsqcup_{i=1}^s S^1\looparrowright\Sigma$ be a multiloop on a surface $\Sigma$. In the unpinning avoidance game, two players alternately remove pins while leaving a pinning set after each turn; the first player with no legal move loses. Let $\mathsf{Unpin_{AVOID}}(\Sigma,s)$ be the decision problem asking whether Player 1 has a winning strategy, and let $\mathsf{SimpleUnpin_{AVOID}}(\Sigma,s)$ be its restriction to simple multiloops.

Unpinning avoidance game complexity conjecture. The problem $\mathsf{Unpin_{AVOID}}(\Sigma,s)$ is $\textsf{PSPACE}$-complete for any $s$ and $\Sigma$, and $\mathsf{SimpleUnpin_{AVOID}}(\Sigma,s)$ is $\textsf{PSPACE}$-complete for any $s\geq 4$.

The conjecture is motivated by the relationship with Node Kayles, which is known to be $\textsf{PSPACE}$-complete, but the paper leaves these game-complexity claims unresolved.