Torsion subgroups of elliptic curves in short Weierstrass form

In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form, subject to certain inequalities for their coefficients. We provide a series of counterexamples to this claim and explore a number of related results. In particular, we show that, for any $\varepsilon>0$, all but finitely many curves

where $A$ and $B$ are integers satisfying $A>\vert B\vert^{1+\varepsilon}>0$, have rational torsion subgroups of order either one or three. If we modify our demands upon the coefficients to $\vert A\vert>\vert B\vert^{2+\varepsilon}>0$, then the $E_{A,B}$ now have trivial rational torsion, with at most finitely many exceptions, at least under the assumption of the abc-conjecture of Masser and Oesterlé.