The Strong Torsion Conjecture for abelian varieties

Let $K$ be a number field, let $g,d\geq 1$, and let $n(g,d)\geq 1$ be an integer depending on $g$ and $d$.

Strong Torsion Conjecture. If $n\geq n(g,d)$, then there are no dimension-$g$ abelian varieties $A$ defined over $K$, with $[K:\mathbb Q]\leq d$, that have a $K$-rational torsion point of order $n$.

The conjecture is the abelian-variety torsion statement used in the source to obtain the weak Main Conjecture for modular towers. The excerpt attributes it to the cited literature but does not supply a resolution.