I really should know the answer to this, but I don't, so I'll ask here.
A Q-curve is an elliptic curve E over Q-bar which is isogenous to all its Galois conjugates. A Q-curve is modular if it's isogenous (over Q-bar) to some factor of the Jacobian of X_1(N) for some N>=1 (here X_1(N) is the compact modular curve over Q-bar).
Has current machinery proved the well-known conjecture that all Q-curves are modular yet?
Remark: I know there are many partial results. What I'm trying to establish is whether things like Khare-Wintenberger plus best-known modularity lifting theorems are strong enough to give the full conjecture yet, or whether we're still waiting.