TITLE: Is it true that irreducible generic representations of $G_2(F)$ are self-dual?
QUESTION [9 upvotes]: Let $G_2$ be the split exceptional group of type $G_2$ and $F$ be a p-adic field. Is it true that every irreducible smooth representation of $G_2(F)$ is self-contragredient? If the answer is Yes, can anybody give me a reference? If not, is there a way to describe the contragredient, for example, is there a MVW involution like the classical group case?
Edit：
I thank Paul Garrett and Jim Humphreys for their comments and Jeffrey Adams for his nice answer. According to Jeffrey Adams's answer, one expects that each L-packet of $G_2(F)$ is self-dual. On the other hand, according to the general philosophy of Gan–Gross–Prasad, in each L-packet, there should be at most one generic representation (GGP conjectured that in each generic local L-packet, there is at most one representation which has the given Bessel model or Fourier–Jacobi model for classical groups. I do not know if anybody conjectured this for exceptional groups. But I just think that we can expect this once we can define the corresponding model. In particular, one would expect the uniqueness of generic member in each L-packet. For the conjecture of uniqueness of generic element in each L-packet, there is probably early reference, but I learned it from GGP). Thus one would expect that:
Each generic smooth irreducible representation of $G_2(F)$ is self-dual.
Do we expect this or is this also false? If we do expect this, do we know anything related to this?

REPLY [6 votes]: $\DeclareMathOperator\PGSp{PGSp}\DeclareMathOperator\PGL{PGL}$Generic representations of a $p$-adic $G_2$ are indeed self dual. It suffices to prove this for super-cuspidal representations. 
Observe that, for unitary representations, taking dual is the same as taking complex conjugate. 
Now all generic super-cuspidal lift one-to-one to generic representations of $\PGSp_6$ by the exceptional theta correspondence, see Savin–Weissman - Dichotomy for generic supercuspidal representations of $G_2$, Compositio Math (2011) (MSN). Since the exceptional theta correspondence commutes with complex conjugation, the statement follows from self-duality on $\PGSp_6$ side. In fact, since any representation of $G_2$ lifts either to $\PGSp_6$ or to a compact form of $\PGL_3$, one can completely classify representations of $G_2$ that are not self dual: the super-cuspidal representations that correspond to non-trivial representations of the compact form of $\PGL_3$.