Conjecture on exceptional pairs for representable prime squares

Let $a_1$ and $a_2$ be integers with $a_2>a_1>40$ and $\gcd(a_1,a_2)=1$. Let $\pi_{2,a_1,a_2}$ denote the number of prime squares $p^2\le g_{a_1,a_2}$ representable as $p^2=a_1x_1+a_2x_2$ with $x_1,x_2\in\mathbb Z_{\ge 0}$. Exceptional-pair conjecture. One has $\pi_{2,a_1,a_2}>0$. Moreover, only finitely many coprime pairs $(a_1,a_2)$ satisfy $\pi_{2,a_1,a_2}=0$, apart from the families $(6,6g+5)$, $(8,8g+7)$, $(12,12g+11)$, and $(24,24g+23)$ identified in the theorem preceding the conjecture. The conjecture is based on numerical experiments, and the text gives no resolution.