On the Kodaira-Spencer's problem on almost Hermitian $4$-manifolds

In 1954, Hirzebruch reported a problem posed by Kodaira and Spencer: on compact almost complex manifolds, is the dimension $h^{p,q}_{\bar \partial}$ of the kernel of the Dolbeault Laplacian independent of the choice of almost Hermitian metric? In this paper, we review recent progresses on the original problem and we introduce a similar one: on compact almost complex manifolds, find a generalization of Bott-Chern and Aeppli numbers which is metric-independent. We find a solution to our problem valid on almost K\"ahler $4$-manifolds.