Kinoshita conjecture on reducibility of knotted projective planes

A smoothly embedded projective plane in $S^4$ is knotted if it is not the standard unknotted embedding, and it is reducible if it admits an unknotted summand that is not a $2$-sphere. Kinoshita's conjecture. Every knotted projective plane in $S^4$ is reducible. The conjecture was posed as a well-known claim in the study of embedded surfaces in $S^4$ and is disproved by the paper's construction of an irreducible embedded projective plane.