Boundary-path correspondence conjecture for compactified open-curve moduli spaces

Let $X$ be the bordered holomorphic curve and let $\tilde M(f)$ be the compactified moduli space associated with a fixed homotopy class $[f]$. A simple path in $X$ has no self-intersections, and its endpoints lie on the boundary away from the marked points. Boundary-path correspondence conjecture. Isotopy classes of such paths, taken up to the action of the mapping class group fixing the marked boundary points, are in one-to-one correspondence with codimension-one components of the boundary of $\tilde M(f)$. This correspondence is intended to describe the codimension-one boundary strata governing the stringy-category relations. The supplied text gives no resolution.