# finding set of tree decompositions to cover all pairs of vertices

Let $P(X_i=x)$ represent probability that randomly chosen proper $q$-coloring of an $L\times L$ square grid contains color $x$ at position $i$. How do you efficiently compute $P(X_{i}=X_{j})$ for every pair $(i,j)$?
Fastest known method for counting $q$-proper colorings on grids uses tree decomposition. To extend it to this problem, one needs a set of tree decompositions so that every pair of vertices is contained in some bag. Is anything known about this problem?