Finiteness conjecture for irreducible 4-manifolds of fixed complexity

Let $c$ be a natural number, and let a complexity-$c$ irreducible 4-manifold mean an irreducible 4-manifold having complexity $c$. Finiteness conjecture. For every natural number $c$, there are only finitely many irreducible 4-manifolds of complexity $c$ that are not doubles of 2-handlebodies. The paper explains that doubles of 2-handlebodies are expected to occur infinitely often for every $c$, so excluding them is essential to this finiteness claim; the conjecture remains open.