The prime-square plus prime conjecture for nonsquares

Let $N>24476$ be a non-square satisfying $N\equiv 0,2\pmod{6}$. Nonsquare prime-square representation conjecture. Then there exist primes $p$ and $q$ such that

$$N=p^2+q.$$

By Hua's theorem, almost all integers in the indicated congruence classes have such a representation, while the conjecture asserts it for every $N$ above the explicit threshold; the source gives no resolution.