Typically bounding torsion on elliptic curves with rational $j$-invariant

A family $mathcal{F}$ of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups $E(F)[$tors$]$ of those elliptic curves $E_{/F}in mathcal{F}$ can be made uniformly bounded after removing from $mathcal{F}$ those whose number field degrees lie in a subset of $mathbb{Z}^+$ with arbitrarily small upper density. For every number field $F$, we prove unconditionally that the family $mathcal{E}_F$ of elliptic curves over number fields with $F$-rational $j$-invariants is typically bounded in torsion. For any integer $dinmathbb{Z}^+$, we also strengthen a result on typically bounding torsion for the family $mathcal{E}_d$ of elliptic curves over number fields with degree $d$ $j$-invariants.