Klein's cup-one length conjecture

Let $X$ be a space and let $n>2$. Assume that $LS_2(X)\leq n-2$, where $LS_2$ is the local LS-category invariant defined using an $n$-set cover whose pairwise unions map null homotopically to $X$. Klein's conjecture. There is a notion of cup-one length such that $LS_2(X)\leq n-2$ implies that the cup-one length of $X$ is at most $n-2$. The proposed invariant is intended to measure higher cup-one structure associated to the local LS-category; it is explicitly described as not yet defined, and no resolution is given.