Kinoshita's problem on reducibility of projective-plane knots

A $P^2$-knot is a surface-knot homeomorphic to the projective plane $\mathbb{R}\mathrm{P}^2$, and a standard $P^2$-knot is the standard projective-plane knot. A surface-link is $P^2$-reducible if it is a connected sum of a standard $P^2$-knot and some surface-link; otherwise it is $P^2$-irreducible.

Kinoshita's problem. Every $P^2$-knot is $P^2$-reducible, that is, every $P^2$-knot is a connected sum of a standard $P^2$-knot and a surface-knot homeomorphic to the 2-sphere $S^2$.

This is presented as one of the fundamental conjectures for $P^2$-knots. The statement asks whether every projective-plane knot splits into a standard projective-plane summand and a spherical-knot summand; its resolution is not indicated here.