This is definitely an open problem.
For a very recent reference of $Sym^m\pi$ being known only for $m \leq 4$ ($\pi$ arbitrary cuspidal), see:
As eric mentions in the comments, arithmetic techniques make some cases like holomorphic forms accesible, but those techniques are certenly missing in this case, $\pi$ a Maass form.
Also, more concretely for Maass forms, see this paragraph on a 2003 Sarnak paper (Spectra of Hyperbolic Surfaces):
This [Kim-Sarnak result towards the eigenvalue conjecture for Maass
forms] is getting close to $1/4$, but it is also close to the limit of
these methods. The functorial lifts $sym^3$ and $sym^4$ are based on
the continuous spectrum (Eisenstein series) on exceptional groups
including $E_8$. What can be done this way terminates with the finite
list of exceptional groups.
This is closely related with what Marty and Garret said in their very interesting comments above.