  Let $F$ be a number field unramified at an odd prime $p$ and $F_\infty$ be
the $\mathbf{Z}_p$-cyclotomic extension of $F$. Let $A$ be an abelian variety
defined over $F$ with good supersingular reduction at all primes of $F$ above
$p$. B\"uy\"ukboduk and the first named author have defined modified Selmer
groups associated to $A$ over $F_\infty$. Assuming that the Pontryagin dual of
these Selmer groups are torsion
$\mathbf{Z}_p[[\mathrm{Gal}(F_\infty/F)]]$-modules, we give an explicit
sufficient condition for the rank of the Mordell-Weil group $A(F_n)$ to be
bounded as $n$ varies.
