Erdős Problem #17 — Primes below which every even number is a difference of primes

A purely computational problem (this problem cannot be attacked by other means at present). Call a prime $p$ good if every even number $2r \leq p - 3$ can be written in the form $q_1 - q_2$ where $q_1 \leq p$, $q_2 \leq p$ are primes. Are there infinitely many good primes?