  Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in
which $p$ splits. Let $A$ be an abelian variety defined over $K$ with
supersingular reduction at both primes above $p$ in $K$. Under certain
assumptions, we give a growth estimate for the Mordell--Weil rank of $A$ over
finite extensions inside the $\mathbb{Z}_p^2$-extension of $K$. In the last
section, written by Chris Williams, he includes some speculative remarks on the
$p$-adic $L$-functions for $\mathrm{GSp}(4)$ corresponding to the multi-signed
Selmer groups constructed in this paper.
