Is it true that irreducible generic representations of $G_2(F)$ are self-dual? 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? 
 A: $\DeclareMathOperator\PGSp{PGSp}\DeclareMathOperator\PGL{PGL}$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 $\PGL_3$. 
A: Since $-1$ is in the Weyl group (over $F$, not just the algebraic closure)  you might expect every irreducible representation to be self-dual. This is the case over $\mathbb R$. It is false over a $p$-adic field, but subtle, and it is not easy to construct an example.
There are non-self dual cuspidal unipotent represenations of $G_2(k)$ where $k$ is the (finite) residue field. By the standard pull back and induction procedure these give rise to non-self-dual supercuspidal representations of $G_2(F)$. The same thing works for all exceptional groups.
Dipendra Prasad has a discussion of some closely related matters  in
A 'relative' local Langlands Correspondence (arXiv:1512:04347).
In terms of involutions, $G_2(F)$ (for example) has an involution $\tau$ such that $\tau(g)$ is conjugate to $g^{-1}$ over the algebraic closure.
This is the "Chevalley involution", and it is inner for $G_2(F)$. However $\tau(g)$ cannot always be $G_2(F)$-conjugate to $g^{-1}$ (exactly because then every irreducible representation would be self-dual, which is false.)
See The Real Chevalley Involution (arXiv:1203:1901), page 4.
Finally, because of the involution just mentioned,
one would expect that every L-packet for $G_2(F)$ is self-dual, but the duality operation could be nontrivial on the packet.
