Let $E$ be an elliptic curve over $\mathbb{Q}$ and $T$ the $p$-adic Tate module of $E$. Kato's Euler system, constructed in the paper "P-adic Hodge theory and values of zeta functions of modular forms" (Asterisque 295, 2004), gives rise to an element $\mathbf{z}_{\rm Kato}$ lying in the Iwasawa cohomology $ H^1_{\mathrm{Iw}}(\mathbb{Q}_p, T)[\frac{1}{p}]$. In Theorem 12.5(4) of the paper, and he Kato shows that if the image of the Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ in $\mathrm{Aut}(T)$ contains $\mathrm{SL}_2(\mathbb{Z}_p)$, then in fact $\mathbf{z}_{\rm Kato} \in H^1_{\mathrm{Iw}}(\mathbb{Q}_p, T)$.
Is it known if there are weaker conditions that are sufficient to ensure that $\mathbf{z}_{\rm Kato}$ has this integrality property? Are there examples where it is genuinely non-integral, or is it conjectured that it should always be so?
I would be the last person to claim I understand Kato's argument, but it looks to me as if he only actually uses the weaker statement that the mod $p$ Galois representation $T/pT$ is irreducible. I'd be interested to know if this weaker condition is indeed sufficient, and whether anything is known in this direction if the weaker condition doesn't hold (i.e. if $E$ admits a $p$-isogeny).
(EDIT: Added more detail and references.)
