Existence of non-strongly p-artinian groups and non-LFS-realizable fusion systems

A group is $p'$-reduced if it has no nontrivial normal $p'$-subgroups. A locally finite group is $p$-artinian if it satisfies the relevant descending-chain condition, and it is strongly $p$-artinian when it also satisfies the additional centralizer condition specified in the paper. A fusion system is LF-realizable if it is realized by a locally finite group, and LFS-realizable if it is realized by a locally finite, strongly $p$-artinian group.

Existence conjecture. There are $p'$-reduced locally finite $p$-artinian groups that are not locally finite, strongly $p$-artinian groups, and there are fusion systems that are LF-realizable but not LFS-realizable.

The conjecture proposes that both separations occur: the class of locally finite $p$-artinian groups is strictly broader than the class of locally finite, strongly $p$-artinian groups, and LF-realizability is strictly broader than LFS-realizability. The source states that no such examples are known and that constructing them appears difficult.