Polynomial torsion bound conjecture for elliptic curves

Let $F$ be a number field of degree $d=[F:\mathbb{Q}]$, and let $E/F$ be an elliptic curve. Write $E(F)[\operatorname{tors}]$ for the subgroup of $F$-rational points of finite order. Polynomial torsion bound conjecture. There exist absolute constants $c,B>0$ such that

$$\#E(F)[\operatorname{tors}]\leq c\cdot d^{B}.$$

Merel's theorem gives a uniform bound depending only on $d$, but the known bounds are more than exponential in the degree. For CM elliptic curves, a bound of order $d\log\log d$ is known; this conjecture asks for a polynomial bound in the non-CM setting as well as in general.