Ulam's greedy prime sequence $q_{n+1}=$ least prime $q_n+q_i-1$: can it be infinite? (Erdős #472)

Start from a finite increasing sequence of primes $q_1<\cdots<q_m$. Extend it greedily: for $n\geq m$, let $q_{n+1}$ be the smallest prime of the form $q_n+q_i-1$ with $1\leq i\leq n$; if no value $q_n+q_i-1$ is prime, the sequence terminates. Is there a choice of initial sequence for which the resulting sequence never terminates — that is, is infinite? For instance, starting from $3,5$ yields $3,5,7,11,13,17,\ldots$.

Acceptance. FULLY RESOLVES (proof-shaped): exhibit a starting sequence together with a proof that its greedy continuation is infinite (a finite proof of infinitude, e.g. an invariant or covering argument guaranteeing an admissible prime always exists); OR prove that every starting sequence terminates. ADVANCES: a reproducible computation extending the sequence from stated seeds to a new record length far beyond current records / OEIS A389713, with the program and the exact terms; or a proof that infinitude holds or fails under a clearly stated conjecture (e.g. a Dickson / Hardy–Littlewood-type hypothesis on primes in the relevant linear forms). Deliver the proof, or the search code plus the record orbit.