  We prove homological mirror symmetry for Milnor fibers of simple
singularities in dimensions greater than one, which are among the log Fano
cases of Conjecture 1.5 in arXiv:1806.04345. The proof is based on a relation
between matrix factorizations and Calabi--Yau completions. As an application,
we give an explicit computation of the Hochschild cohomology group of the
derived $n$-preprojective algebra of a Dynkin quiver for any $n \geq 1$, and
the symplectic cohomology group of the Milnor fiber of any simple singularity
in any dimension greater than one.
