Chebotarev–Sato–Tate square-root error conjecture for certain abelian surfaces

Let $A/\mathbb{Q}$ be an abelian surface isogenous to a product of elliptic curves that are not $\overline{\mathbb{Q}}$-isogenous and are non-CM. Let $a_{1,p}$ be the relevant Frobenius trace, let $\mathscr{C}(m,t)$ and $\mathscr{G}(m)$ be the sets and groups appearing in the Chebotarev–Sato–Tate distribution, let $\Phi$ be its density, and define

$$E(x,t,m,I):=\frac{\#\{p\le x: \frac{a_{1,p}}{4\sqrt{p}}\in I,\; a_{1,p}\equiv t\pmod m\}}{\pi(x)}-\frac{\#\mathscr{C}(m,t)}{\#\mathscr{G}(m)}\int_I\Phi(s)\,\mathrm{d}s.$$

Square-root error conjecture. For fixed $t$, $m$, and $I\subset[-1,1]$,

$$E(x,t,m,I)=O(x^{-1/2+\varepsilon})\quad\text{as }x\to\infty$$

for every $\varepsilon>0$. This is the proposed abelian-surface analogue of the Akiyama–Tanigawa conjecture; the supplied text presents it as plausible and gives no resolution.