Refined conjectures on Fitting ideals of Selmer groups over ${\mathbf{Z}}_p^2$-extensions

Let $p>3$ be a prime number and $K$ be an imaginary quadratic field where $p$ splits. Let $K_{\mathrm{\infty }}$ be the ${\mathbf{Z}}_p^2$-extension of $K$ and let $K_n$ be a finite subextension of $K_{\mathrm{\infty }}\mathrm{/}K$. Let $E$ be an elliptic curve with good ordinary reduction at $p$. Under some hypotheses, we show that the Mazur-Tate element attached to $E$ over $K_n$ by S. Haran generates the Fitting ideal of the dual Selmer group of $E$ over $K_n$.