A $\mathrm{GL}_3$ level analog of Selberg's result for $S(t)$

Let $π$ be a spherical cuspidal $\mathrm{GL}_3$ newform of prime level $N$ and trivial central character, and define $S(t,π)=π^{-1}\arg L(1/2+\textrm{i}t,π)$. Assuming the generalized Riemann hypothesis for $L(s,π)$, we establish an asymptotic formula for each fixed weighted moment of $S(t,π)$ as $N