Let $K$ be a finite extension of $\mathbb{Q}$ of degree $d$ and let $E(K)$ be an elliptic curve over the field $K$ with coefficients in $K$. Let us fix $d$ and vary over all the possible $K$, in turn varying over all possible $E$. My question is the following: does there exist an integer $N=N_d$ such that $N \geq \#|E(K_{\mathrm{tors}})|$ for any $K, E$ and fixed $d$? If this is not known to be the case, then what is the general consensus regarding the number of $K$-rational torsion points of $E$? If none of this is known, is there a record for the largest known torsion order? As of right now I'm curious for $d=2$ and non-CM elliptic curves $E(K)$, but any information regarding a specific $d > 1$ is helpful.

Inventiones mathematicæin 1996. For $d=2$ The problem is completely solved. en.wikipedia.org/wiki/Torsion_conjecture $\endgroup$