The hyperbolic L-space conjecture for universally tight contact structures

Let $M$ be a hyperbolic $3$-manifold. An L-space is a rational homology sphere whose first singular homology group has order equal to the free rank of its Heegaard Floer homology group. A contact structure on $M$ is universally tight if its pullback to the universal cover is tight.

Hyperbolic L-space conjecture. $M$ is an L-space if and only if it does not support a universally tight contact structure.

This is proposed as a contact-topological version of the L-space conjecture. Although some non-hyperbolic L-spaces, such as lens spaces, support universally tight contact structures, whether any hyperbolic L-space does so was not known in the stated context.