Legendre-transform equivalence for chord-diagram moduli spaces

Let $M$ be the closed manifold, let $\Gamma$ be a marked chord diagram, let $\sigma$ be an $LM$-Morse structure on $\Gamma$, and let $\epsilon>0$. Write $\mathcal{M}^{\sigma}_\Gamma(LM)$ for the moduli space of the corresponding Morse-theoretic configurations on $LM$, and let $\mathcal{M}^{hol}_{(\Gamma,\sigma,\epsilon)}(T^*M)$ be the moduli space of cylindrical holomorphic curves in $T^*M$.

Legendre-transform equivalence conjecture. The natural map induced by the Legendre transform,

$$\ell:\mathcal{M}^{\sigma}_\Gamma(LM)\to\mathcal{M}^{hol}_{(\Gamma,\sigma,\epsilon)}(T^*M),$$

is a homotopy equivalence for $\epsilon$ sufficiently small.

This would generalize the Salamon–Weber correspondence between gradient trajectories of the Floer action and of the energy functional. It is presented as a possible approach to identifying the Floer-theoretic operations with string topology operations, and no proof or resolution is supplied in the source.