Minimal content conjecture for $p$-adic $L$-functions in Hida families

Let $\bT_{\mathfrak{m}}$ be the relevant ordinary Hecke algebra, let $\mathcal{I}_{\mathfrak{m}}$ be its congruence ideal, and let $L_p^+(\mathfrak{m},\omega^j)$ be the associated $p$-adic $L$-function. Let $\bT_{\mathfrak{m}}^0$ be the corresponding reduced Hecke algebra, and let $H^1_\Lambda(X_1(N))^-_{\mathfrak{m}}$ be the indicated cohomology module. Minimal content conjecture. (1) If $\bT_{\mathfrak{m}}$ is Gorenstein, then for every even integer $j$,

$$\operatorname{content}_{\bT_{\mathfrak{m}}}(L_p^+(\mathfrak{m},\omega^j))=\mathcal{I}_{\mathfrak{m}}.$$

(2) If $\bT_{\mathfrak{m}}^0$ is Gorenstein and $e$ generates $H^1_\Lambda(X_1(N))^-_{\mathfrak{m}}$ as a $\bT_{\mathfrak{m}}^0$-module, then for every even integer $j$, $L_p^+(\mathfrak{m},\omega^j)^0_e$ has unit content. The conjecture predicts that the divisibility forced by the congruence ideal is exactly the full content of the $p$-adic $L$-function, and that the normalized cohomological $p$-adic $L$-function has no further common divisor. The supplied text gives no evidence of resolution.