The characterization conjecture for local LS-category and cocategory

Let $X$ be a space and let $n>2$. Define $LS_2(X)\leq n-2$ when $X$ has Lusternik–Schnirelmann category at most $n$ and admits an open cover $U_1,\ldots,U_n$ such that each map $U_i\cup U_j\to X$ is null homotopic for distinct $i,j$. Dually, define $LS^{op}_2(X)\leq n-2$ when $X$ has Lusternik–Schnirelmann cocategory at most $n$ and satisfies the corresponding co-cover condition. Let $(\mathrm{T}^n)^2\mathbb{I}(X)$ and $\mathrm{T}_n^2\mathbb{I}(X)$ denote the second iterates of the covariant and contravariant approximation constructions on the identity functor. Characterization conjecture. The following equivalences should hold: (1) $X$ has $LS_2(X)\leq n-2$ if and only if the map $(\mathrm{T}^n)^2\mathbb{I}(X)\to X$ has a section; (2) $X$ has $LS^{op}_2(X)\leq n-2$ if and only if $X$ is a retract of $\mathrm{T}_n^2\mathbb{I}(X)$. These would characterize the local LS invariants through the second stages of the dual Goodwillie constructions; the source presents them as a conjectural characterization and gives no resolution.