  We show that the minimal volume entropy of closed manifolds remains
unaffected when nonessential manifolds are added in a connected sum. We combine
this result with the stable cohomotopy invariant of Bauer-Furuta in order to
present an infinite family of four-manifolds with the following properties:
  1) They have positive minimal volume entropy.
  2) They satisfy a strict version of the Gromov-Hitchin-Thorpe inequality,
with a minimal volume entropy term.
  3) They nevertheless admit infinitely many distinct smooth structures for
which no compatible Einstein metric exists.
