Hindry–Silverman bound for the full torsion subgroup

Prove or disprove the existence of an absolute constant c such that every non-CM elliptic curve E defined over a number field F of degree at least 3 satisfies $\#E(F)_{tors} \leq c\sqrt{[F:Q]\log\log [F:Q]}$.

Background

The paper discusses a stronger uniform torsion estimate proposed by Hindry and Silverman. This estimate improves the linear-in-degree bound for the size of the torsion subgroup suggested elsewhere in the paper and is essentially optimal in its dependence on the degree, based on a lower-bound construction cited by the authors.

The paper proves that an analogous bound for the torsion exponent would imply generalized Serre uniformity, but it does not resolve the full-torsion question itself.