Ash–Doud conjecture on Galois representations and eigenclasses in the homology of $\operatorname{GL}(3,\mathbb{Z})$

Let $p$ be a prime and let $\bar{\mathbb{F}}_p$ be an algebraic closure of $\mathbb{F}_p$. Let $\rho:G_{\mathbb{Q}}\to\operatorname{GL}(3,\bar{\mathbb{F}}_p)$ be a Galois representation that is a sum of an irreducible odd two-dimensional representation and a character. Let $N$ be its Serre conductor and let $\epsilon$ be its nebentype. Write $\rho=\sigma\oplus\omega^c\psi$, where $\psi$ has conductor prime to $p$, and let $V$ be an irreducible admissible $\mathbb{F}_p[\operatorname{GL}(3,\mathbb{F}_p)]$-module. For the specified possibilities for the restriction of $\sigma$ to inertia at $p$, define $V$ as follows: if

$$\sigma|_{I_p}\sim\begin{pmatrix}\omega^a&*\\0&\omega^b\end{pmatrix},$$

choose $a,b,c$ modulo $p-1$ with $0<a-b,b-c\leq p$ and $0\leq c<p-1$, imposing $a-b=p$ when $\sigma$ is très ramifiée, and take $V=F(a-2,b-1,c)$; alternatively, choose $a,b,c$ with $0<c-a,a-b\leq p$ and $0\leq b<p-1$, again imposing $a-b=p$ when $\sigma$ is très ramifiée, and take $V=F(c-2,a-1,b)$. If

$$\sigma|_{I_p}\sim\begin{pmatrix}\omega_2^{a+bp}&0\\0&\omega_2'^{a+bp}\end{pmatrix}$$

and $0<a-b\leq p$, either choose $0<a-b,b-c\leq p$, $0\leq c<p-1$, and take $V=F(a-2,b-1,c)$, or choose $0<c-a,a-b\leq p$, $0\leq b<p-1$, and take $V=F(c-2,a-1,b)$. Ash–Doud conjecture. For every such value of $V$, there is an $\mathcal{H}_{3,N}$-eigenclass in

$$H_3(\Gamma_0(3,N),V\otimes\epsilon)$$

with $\rho$ attached. This predicts that the specified three-dimensional Galois representations occur in the appropriate arithmetic homology with precisely the Serre weights determined by their inertia at $p$; the broader framework is related to conjectures for more general representations, while the status of this formulation is not resolved in the supplied source.