Conjecture that the skeleton is relative symplectic-cohomology full

Let $a\in QH^0(M;\Lambda)$ be the idempotent generating the ideal $QH^*(M;\Lambda)_{\mathrm{crit}}$, and let $\mathbb{L}$ be the skeleton. A subset is $a$-$SH$-full when its $a$-relative symplectic cohomology detects the corresponding relative theory. Skeleton fullness conjecture. Under the same hypotheses as Theorem

$$, without assuming Hypothesis$$

, the skeleton $\mathbb{L}$ is $a$-$SH$-full. This would extend the fullness result to the non-graded-bounded case and implies intersection consequences for $a$-Floer-theoretically essential monotone Lagrangians; it remains open.