  In this paper, we use homological techniques and a theorem of Thurston to
study homological finiteness of $\text{BDiff}(M, \text{rel }\partial)$ when $M$
is a reducible $3$-manifold with a non-empty boundary that has distinct
irreducible factors. Kontsevich conjectured that $\text{BDiff}(M, \text{rel
}\partial)$ has the homotopy type of a finite CW complex for all $3$-manifolds
with non-empty boundary. Hatcher-McCullough proved this conjecture when $M$ is
irreducible. We show that when $M$ is reducible with non-empty boundary that
has distinct irreducible factors, $\text{BDiff}(M, \text{rel }\partial)$ is
homology isomorphic to a finite CW-complex.
