Non-principality of supersingular cyclotomic Fitting ideals

Determine whether the initial Fitting ideal of the Pontryagin dual of the Selmer group (Sel(\mathbb{Q}_n,E[p^\infty])^\vee) over the finite cyclotomic group ring \(\Lambda_n=\mathbb{Z}_p[\operatorname{Gal}(\mathbb{Q}_n/\mathbb{Q})]\) is non-principal when the elliptic curve \(E/\mathbb{Q}\) has supersingular reduction at \(p\).

Background

In the cyclotomic setting, the paper explains that for an elliptic curve with good ordinary reduction at $p$, the finite-layer Fitting ideal of the Pontryagin dual of the Selmer group is principal and is generated by the relevant finite-layer $p$-adic $L$-function or Mazur–Tate data. The authors then contrast this with the supersingular case, where the corresponding finite-layer Fitting ideal is expected to have a different, non-principal structure. Establishing this non-principality would clarify how supersingular reduction changes the refined finite-layer structure in cyclotomic Iwasawa theory.