Distance-matrix complement conjecture for coalescing cospectral sets

Let $H_1$ and $H_2$ be graphs, and let $B_1\subseteq V(H_1)$ and $B_2\subseteq V(H_2)$ be vertex sets. Let $\mathcal{D}$ denote the distance matrix, whose $(u,v)$-entry records the distance between vertices $u$ and $v$. Distance-matrix coalescing conjecture. If $(H_1,B_1)$ and $(H_2,B_2)$ are coalescing cospectral with respect to $\mathcal{D}$, then $(H_1,V(H_1)\setminus B_1)$ and $(H_2,V(H_2)\setminus B_2)$ are also coalescing cospectral with respect to $\mathcal{D}$. This conjecture is motivated by computational experimentation on small graphs and would extend the complement result proved in the paper to the distance matrix. An obstacle to proving it is that the authors' techniques rely on cycle decompositions to express characteristic polynomials of coalesced graphs in simple terms.