Let $G=Sp_4$ over a p-adic field $F$. After Gan-Takeda's work, the local Langlands correspondence for $G$ is known. Thus we have the local functoriality from $G$ to $GL_5$. Do we know that this local Langlands correspondence preserve the genericity?

More precisely, let $\pi$ be an irreducible generic representation of $Sp_4(F)$, and $\phi: W_F'\rightarrow SO_5(\textbf{C})$ be the corresponding Langlands parameter determined by Gan-Takeda. Consider the Langlands parameter $\phi': W_F'\rightarrow SO_5(\textbf{C})\rightarrow GL_5(\textbf{C})$ by composing $\phi$ with the embedding of $SO_5$ into $GL_5$. Let $\pi'$ be the representation of $GL_5(F)$ corresponding to $\phi'$ by the local Langlands correspondence for $GL$. Do we know $\pi'$ is generic?

This question makes sense for general reductive group $G$ once we know the local Langlands correspondence for $G$. What is the general picture (or conjecture) for a general group $G$? Thanks in advance.