Half-of-primes conjecture for primes outside a two-generated numerical semigroup

Let $a$ and $b$ be integers with $b>a\ge 1$ and $\gcd(a,b)=1$. Set $S(a,b)=ab-a-b$, and let $\pi^*(a,b)$ be the number of primes not belonging to the numerical semigroup $\langle a,b\rangle$; let $\pi(x)$ denote the number of primes not exceeding $x$. Half-of-primes conjecture. One has

$$\pi^*(a,b)\ge \frac 12\pi(S(a,b)),$$

with equality if and only if one of the following holds: (1) $a=1$; (2) $a=2$ and $b=3$; (3) $a=2$ and $b=5$; (4) $a=3$ and $b=5$. The paper proves a weaker universal bound with constant $0.04$ and notes that this constant can be improved; the displayed inequality and its equality classification remain the proposed conjecture.