Serre's companion form conjecture for tame ramification of modular Galois representations

Let $f=\sum a_nq^n$ be an ordinary newform of type $(k,\epsilon)$ for $\Gamma_1(N)$, defined over a field $E$ of characteristic $p$, with $(p,N)=1$, and let

$$\rho_{f,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow \operatorname{GL}_2(E)$$

be the Galois representation attached to $f$. Write $\theta$ for the operator occurring in the companion-form relation, and set $k'=p+1-k$. Serre's conjecture. If $2\leq k\leq p$, then $\rho_{f,p}$ is tamely ramified when $k\neq p$, and unramified when $k=p$, if and only if there exists an eigenform $g=\sum b_nq^n$ of type $(k',\epsilon)$ satisfying

$$\theta g=\theta^{k'}f.$$

Such a form $g$ is called a companion form. This conjecture gives a modular criterion for the ramification behavior at $p$ of Galois representations attached to ordinary exceptional modular forms; the supplied text does not state whether the criterion has been proved or disproved.