Phantom-class conjecture for finite-flat Eisenstein extensions

Let $\mathfrak{q}$ be a prime ideal with $N(\mathfrak{q})=q\not\equiv1\pmod p$. Suppose that the finite-flat extension group $\operatorname{Ext}^1(\mu_p,\mathbf{Z}/p\mathbf{Z})$ over $\operatorname{Spec}\mathscr{O}_F[1/\mathfrak{q}]$ is nonzero, while the corresponding extension group over $\operatorname{Spec}\mathscr{O}_F$ is trivial. Phantom-class conjecture. For every auxiliary prime ideal $\mathfrak{r}$, there exists a cyclotomic-Eisenstein maximal ideal $\mathfrak{m}$ of the Hecke algebra $\mathbf{T}_{\mathfrak{q}\mathfrak{r}}$ of residue characteristic $p$, but no such ideal exists for $\mathbf{T}_{\mathfrak{q}}$. This predicts that the extension is detected only after adding an auxiliary level, producing a phantom class at level $\mathfrak{q}$.