The Iwasawa distinguished-polynomial square-freeness and order conjecture

Let $k$ be a totally real number field, let $p$ be a prime, let $E$ be a finite extension of $\mathbb Q_p$, and let $\chi:G_k\to E$ be an even Artin character realizable over $E$. For one-dimensional $\chi$, let $M(\chi)=\mathcal O_E$ with $G_k$ acting through $\chi$. Assume that $S$ contains the primes above $p$ and the finite primes where $k_\chi/k$ is ramified. Let $\mathcal M$ be the Galois group of the maximal Abelian extension of $(k_\chi)_\infty$ unramified outside the primes above $p$ and infinity, and set $Y=\mathcal M\otimes_{\mathbb Z_p[G]}M(\chi)$. Iwasawa distinguished-polynomial conjecture. For $e\in\mathfrak B_k(E)$, the following assertions are conjectured: (1) if $\chi$ is one-dimensional of order prime to $p$, the distinguished polynomials $g_j$ corresponding to $Y$ are square-free; (2) under the same hypothesis, these $g_j$ have no multiple root $z\in E$ satisfying $|z-1|_p<p^{-1/(p-1)}$; (3) under the same hypothesis, equality holds in

$$\nu_S(1-e,\chi)=\operatorname{cork}_{2,S}(1-e,\chi)-\operatorname{cork}_{0,S}(1-e,\chi);$$

(4) equality holds in this formula without the one-dimensional hypothesis. These claims are presented as partly folklore conjectures motivated by the structure of the relevant Iwasawa modules; the source does not report a resolution.