The cord algebra conjecture for tangle contact homology

Let $T$ be a framed oriented $r$-component tangle in ${\mathbb{R}}^3_{x\geq 0}$, let $N(\varLambda_T)$ be its specified neighborhood, and let $H$ denote the reference subspace used to define the relative framed cord algebra $\operatorname{Cord}(T,H)$. For meridians $\mu_1,\ldots,\mu_r$, write $KCH^0(T,h)$ for degree-zero knot contact homology associated to a handle decomposition $h$ of $N(\varLambda_T)$. The cord algebra conjecture. There exists a choice of handle decomposition $h$ of $N(\varLambda_T)$ such that there is an isomorphism of $\mathbb{Z}[\mu_1^{\pm 1},\ldots,\mu_r^{\pm 1}]$-algebras

$$KCH^0(T,h)\cong \operatorname{Cord}(T,H).$$

This conjecture extends the relationship between knot contact homology and cord algebras to tangles, allowing cords to have either or both endpoints on $H$. The source gives no resolution, so the conjecture is recorded as open; it also notes a further conjectural refinement concerning a handle decomposition with a single top handle.