Consider an irreducible $\mathrm{mod}$ $p$ representation:

$$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$

If $\rho$ is odd, it was conjectured by Serre in the 70s and proved by Khare-Wintenberger in the 00s that $\rho$ comes from a modular form.

But much like in the characteristic 0 case, for $\rho$ even the situation is even more complicated. In particular, it can be easily proved that they never arise form modular forms.

The question is, where are (irreducible) even mod p Galois representations conjectured to come from? What is the analogue of Serre's conjecture in the even case?

This issue has been mentioned several times on MO. For example, this 2012 question:

Is there a theory of Maaß forms over finite fields?, by Chandan Singh Dalawat.

asks something a bit weaker, whether those representations come from Maass forms, for which I'm quite sure the answer is no, but I wouldn't know how to justify it.

Kevin Buzzard briefly addressed this issue in his reply to this 2010 question:

Is there a canonical notion of “mod-l automorphic representation”?, by David Hansen.

And Jim Stankewicz in his answer to this other question:

The significance of modularity for all Galois representations, by Jonah Sinick

although it seems to be some confusion with mod *l* representations, see the comments.

The closest to an answer I've come across is this blog post, which seems to suggest that even representations are either dihedral or automorphic from $U(3)$, but I don't quite follow the argument.

Any information or reference would be appreciated.