For large $n$, must the greedy $[1,n)$ sequence with all prime factors $>n-a$ include a composite? (Erdős #430)

Fix an integer $n$ and define a strictly decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\ge 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1})$ all of whose prime factors are $>n-a_k$ (note the admissibility condition on $a_k$ depends on $a_k$ itself). The sequence terminates when no such integer exists. For example, when $n=8$ one gets $a_1=7$, $a_2=5$, and then it stops. Is it true that for all sufficiently large $n$ not every term of this sequence is prime — equivalently, that the sequence must contain at least one composite term?

Acceptance. FULLY RESOLVES: a proof that for all sufficiently large $n$ the sequence contains a composite term — a complete proof (e.g. via a rigorous reduction to Erdős #385), machine-checkable or fully written — OR a disproof exhibiting the mechanism behind infinitely many $n$ whose entire sequence is prime. ADVANCES (each independently checkable): (a) verify computationally, with reproducible code and a certificate of exhaustiveness, that every $n$ in a stated range $[n_0,N]$ (for a record $N$ beyond anything previously checked) yields a sequence containing a composite; (b) prove, for all large $n$, existence of the required composite under a clearly stated hypothesis, or rigorously establish the equivalence with (the first part of) Erdős #385. Deliver the proof, or the verification code plus the attained range and its certificate.