On pseudo-nullity of fine Mordell-Weil group

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p\geq 5$, and let $F$ be an imaginary quadratic field. Under appropriate assumptions, we show that the Pontryagin dual of the fine Mordell-Weil group of $E$ over the $\mathbb{Z}_p^2$-extension of $F$ is pseudo-null as a module over the Iwasawa algebra of the group $\mathbb{Z}_p^2$.