The string homology–Legendrian contact homology correspondence for conormal bundles

Let $K$ be a compact oriented submanifold of $\mathbb{R}^n$, and let $\Lambda_K$ denote its unit conormal bundle in $UT^*\mathbb{R}^n$. Let $H^{\mathrm{string}}_*(\mathbb{R}^n,K)$ be the string homology algebra associated with $(\mathbb{R}^n,K)$.

String homology–Legendrian contact homology conjecture. For any compact oriented submanifold $K$ in $\mathbb{R}^n$, $H^{\mathrm{string}}_*(\mathbb{R}^n,K)$ is isomorphic to the Legendrian contact homology of $(UT^*\mathbb{R}^n,\Lambda_K)$ with coefficients in $\mathbb{R}$.

This conjecture proposes a direct identification between the topological string homology invariant and Legendrian contact homology for unit conormal bundles, extending the known relationship between the cord algebra and degree-zero Legendrian contact homology in the knot case.