  In this paper, we use homological techniques to study the homological
finiteness of $\mathrm{BDiff}(M, \text{rel }\partial)$ when $M$ is a reducible
$3$-manifold with a non-empty boundary. Kontsevich conjectured that
$\mathrm{BDiff}(M, \text{rel }\partial)$ 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 prove the homological
version of Kontsevich's conjecture which is to show that $\mathrm{BDiff}(M,
\text{rel }\partial)$ has finitely many nonzero homology groups which are each
finitely generated.
