Commutativity conjecture for iterated vanishing-cycle correspondences

Let $\Sigma$ be a surface and let $L,L'\subset\Sigma$ be disjoint circles. Write $\widehat{V}_L^n$ and $\widehat{V}_{L'}^n$ for the associated Lagrangian correspondences between the relevant symmetric products, and use $\circ$ for correspondence composition. Commutativity conjecture. The two iterated vanishing-cycle correspondences are Hamiltonian isotopic:

$$\widehat{V}_{L'}^{n-1}\circ\widehat{V}_L^n\simeq_H\widehat{V}_L^{n-1}\circ\widehat{V}_{L'}^n.$$

This would strengthen the preceding smooth-isotopy statement to Hamiltonian isotopy and is the commutativity property needed to establish independence of perturbations in the construction of the Lagrangian matching invariant; the supplied text gives no resolution status.