State-certification separation under joint full-rank and projectivity assumptions

Determine whether there exists a nonlocal game that self-tests a full-rank projective entangled reference strategy within the class of pure full-Schmidt-rank projective strategies while admitting an unrestricted optimal strategy from which the reference entangled state cannot be extracted.

Background

The paper constructs a game separating full-rank projective self-testing from assumption-free self-testing at the level of the complete strategy: every restricted optimum is locally dilatable to the reference strategy, whereas an unrestricted optimum with a nonprojective measurement is inequivalent. However, the unrestricted optimum still permits extraction of the same maximally entangled Bell state. The authors therefore leave unresolved whether the joint full-Schmidt-rank and projectivity assumptions can instead produce a separation at the level of state certification, with the unrestricted optimum failing to yield the reference entangled state itself.