Kulikov's nondecomposability conjecture for transcendental Hodge structures of surfaces

Let $S$ be a smooth projective surface over the complex numbers. Write $T_S$ for the integral polarized Hodge structure on the transcendental part of $H^2(S,\mathbb Z)$. Nondecomposability conjecture. The integral polarized Hodge structure $T_S$ is indecomposable. This conjecture is motivated by the proposed approach to proving irrationality of very general cubic fourfolds: a birational parametrization would force the transcendental Hodge structure of a surface to contain the transcendental structure of the cubic fourfold as a proper summand. The conjecture remains open in the source.