Infinitely many primes $p$ with every $p-k!$ composite (for $k!<p$)? (Erdős #1059)

Is it true that there are infinitely many primes $p$ such that $p-k!$ is composite for every integer $k\geq 1$ with $1\leq k!<p$? Equivalently, are there infinitely many primes $p$ for which none of $p-1!,\,p-2!,\,p-3!,\ldots$ — taken over all factorials $k!$ below $p$ — is itself prime?

Acceptance. FULLY RESOLVES: a complete proof (machine-checkable preferred, else full written proof) that infinitely many primes $p$ have $p-k!$ composite for all $k$ with $k!<p$; a proof of Erdős's suggested easier variant (infinitely many $n\in(l!,(l+1)!]$ with all prime factors $>l$ and all $n-k!$ composite) also fully resolves the stated aim if clearly delimited. ADVANCES: substantially extend the list of primes with this property beyond the known small examples, with the search program and a reproducible certificate that each listed $p$ is prime and that every $p-k!$ with $k!<p$ is composite, establishing a new record search height; or prove a positive-density / quantitative lower bound on the count of such primes up to $x$. Deliver the proof, or the search code plus the extended list and its verification certificate.