The stabilization conjecture for the monopole Floer Legendrian invariant

Let $\mathcal{K}\subset Y$ be a Legendrian knot. Let $S_{+}(\mathcal{K})$ and $S_{-}(\mathcal{K})$ denote its positive and negative stabilizations, respectively, and let $\ell(\mathcal{K})$ be its monopole Floer Legendrian invariant. Stabilization conjecture. One has

$$\ell(S_{-}(\mathcal{K}))=\ell(\mathcal{K})$$

and

$$\ell(S_{+}(\mathcal{K}))=0.$$

This predicts that the invariant is unchanged by negative stabilization and vanishes after positive stabilization, paralleling the expected behavior of established Legendrian invariants. The source gives no resolution of the conjecture.