Cord-ring conjecture for unit conormal contact homology

Let $K$ be a codimension $2$ submanifold of a manifold $M$, let $LK$ be its unit conormal Legendrian submanifold in the contact manifold $ST^*M$ of unit covectors, and let the cord ring of $K$ be the quotient defined from its cord algebra. Cord-ring conjecture. The cord ring of $K$ is the zero-dimensional relative contact homology of $LK$ in $ST^*M$. The cord ring is already an isotopy invariant and distinguishes the unknotted $S^2$ in $S^4$ from spun knots arising from knots with nontrivial cord ring, but the asserted identification with relative contact homology is not established here.