Finiteness conjecture for irreducible simply connected 4-manifolds of fixed complexity

Let $c$ be a natural number, and consider irreducible simply connected $4$-manifolds of complexity $c$. Finiteness conjecture. For every natural number $c$ there are finitely many irreducible simply connected $4$-manifolds of complexity $c$. This is an alternative to the preceding finiteness conjecture after allowing doubles of $2$-handlebodies; the paper proposes it in the context of the filtration by shadow complexity, but does not provide a resolution.