The infinitude conjecture for prime products in $E^*_{Y^2}$

Let $Y$ be an integer, and let $E^*_{Y^2}$ denote the set introduced in the surrounding discussion. Let $a_1,a_2,b_1,b_2$ be odd integers, and let $\pi_1$ and $\pi_2$ be primes satisfying

$$\pi_1=4a_1^2+b_2^2,\qquad \pi_2=a_2^2+4b_1^2.$$

Infinitude conjecture. $E^*_{Y^2}$ contains infinitely many numbers of the form $\pi_1\pi_2$ for such primes $\pi_1$ and $\pi_2$.

The conjecture is proposed as a consequence that would imply the infinitude of $E^2_{Y^2}\mathbin{\diagdown}E^1_{Y^2}$, complementing the observation that $E^1_{Y^2}\mathbin{\diagdown}E^2_{Y^2}$ is infinite. The supplied text gives no resolution, so its status remains open.