Constant-wide Demi-Quantum Nim hardness conjecture

Let $Z=(B,\Sigma,\rho_L,\rho_R,b_0)$ be a combinatorial game instance, and let $\mathbb{B}=[b_1|\cdots|b_s]$ be a quantum board consisting of a superposition of distinct positions. The associated demi-quantum game $Z^{\mathrm{demi}\mathcal Q}[\mathbb{B}]$ starts from $\mathbb{B}$, permits only classical moves, and is called constant-wide when $s$ is constant. Playing optimally means determining the outcome under optimal play.

Constant-wide Demi-Quantum Nim conjecture. Constant-wide $\ruleset{Demi-Quantum Nim}$ is $\cclass{NP}$-hard to play optimally.

The paper establishes that poly-wide Demi-Quantum Nim is $\cclass{PSPACE}$-complete, while the constant-wide hardness claim remains open.