The NLS-detection gluing conjecture for graph manifolds

Let $W$ be a rational homology $3$-sphere graph manifold with JSJ tori, and let $m$ be the number of these tori. For each $i$, let $[\alpha_i]$ be a rational slope associated to the corresponding JSJ torus, and write $[\alpha_*]=([\alpha_1],[\alpha_2],\ldots,[\alpha_m])$. The notation $[\alpha_*^{(i)}]$ denotes the slope data induced on the $i$th piece, and NLS detection means the detection property defined earlier in the source. NLS-detection gluing conjecture. The manifold $W$ is not an L-space if and only if there is an $m$-tuple of rational slopes $[\alpha_*]$ such that, for every $i$, $[\alpha_*^{(i)}]$ is NLS detected. This would provide a Heegaard–Floer gluing theorem for rational homology $3$-sphere graph manifolds and would reduce the cited L-space conjecture to the non-left-orderable case. The source presents this as a conjecture; its notation and the precise definition of NLS detection depend on the preceding sections.