Joint equidistribution of newforms

Let $Y_1$ be a compact arithmetic hyperbolic surface associated to a maximal quaternion order, let $Y_q$ be a cover associated to an Eichler suborder of prime level $q$, and let ${\iota }_q$ be embedding of $Y_q$ as the Hecke correspondence into $Y_1\times Y_1$. Let ${\mu }_1$ and ${\mu }_q$ be the invariant probability measures on $Y_1$ and $Y_q$, respectively. If $F$ is a newform on $Y_q$, we conjecture that the pushforward measure $({\iota }_q{)}_{\ast }(\mid F{\mid }^2{\mu }_q)$ converges weakly to the uniform measure ${\mu }_1\times {\mu }_1$, as $q$ tends to infinity. We prove this conjecture with an effective rate of equidistribution, assuming GRH.