Conjecture on the growth of the prime-power threshold

Let $g(k)$ be the least positive integer such that for every pair of integers satisfying $g(k)<a_1<a_2$, there is a prime power $p^k\le g_{a_1,a_2}$ of the form $p^k=a_1x+a_2y$ with $x,y\in\mathbb Z_{\ge 0}$. Growth conjecture for $g(k)$. For every given real number $M>0$, one has

$$g(k)>M^k$$

for all sufficiently large $k$. The conjecture concerns the asymptotic growth of the threshold beyond which every two-generator Frobenius problem contains a representable $k$th prime power; the text supplies no resolution.