Shevelev–Moses conjecture on merging prime-preserving sequences

Let $H_a$ be the lexicographically first strictly increasing sequence beginning with $H_a(2)=a$ such that $H_a(n)$ is prime if and only if $n$ is prime. For primes $a>b\geq 3$, write $H_a$ and $H_b$ for the sequences with initial terms $a$ and $b$, respectively. Shevelev–Moses conjecture. For arbitrary primes $a>b\geq 3$, the sequences $H_a$ and $H_b$ merge, meaning that they agree from some position onward. This conjecture concerns the interaction between lexicographically first prime-preserving sequences generated from different initial primes; the source provides no resolution, so the conjecture remains open.