The Floer-theoretic identification conjecture for nearby Lagrangian torsion

Let $L\subset T^*S^{4k}$ be a Lagrangian homotopy sphere. Suppose that Floer-theoretic nearby Lagrangian torsion defines a coset $\nu^{\mathrm{FL}}(L)\in\mathbf{R}/\lambda_k\mathbf{Z}$, and let $\nu(L)$ denote the nearby Lagrangian torsion defined in the source.

Floer-theoretic identification conjecture. The Floer-theoretic coset is the same as the nearby Lagrangian torsion:

$$\nu^{\mathrm{FL}}(L)=\nu(L).$$

This identification would relate the proposed Floer-theoretic construction to the existing nearby Lagrangian torsion and is presented by the source as an unresolved conjectural equality.