Exclusion of the infinite chain condition for universal covers of surfaces

Let $S$ be a smooth projective surface, and let $\widetilde{S}$ be its universal covering space. Infinite-chain conjecture. The infinite chain condition can be ruled out for $\widetilde{S}$. The claim is presented as a consequence of the Stability Hodge Structure considerations and is attributed in the source to EKPR; the source gives no resolution beyond this assertion.