Infinitely many wisde primes

Prove that infinitely many primes p satisfy δ(p)=⌊(p/2)^{1/3}⌋, equivalently, prove that at least one of the nine explicit cubic polynomials P_1,…,P_9 takes prime values infinitely often.

Background

A prime is called wisde when the upper bound δ(p)≤⌊(p/2)^{1/3}⌋ is attained. The paper classifies all sufficiently large wisde primes using nine explicit cubic polynomials P_1,…,P_9: a sufficiently large prime is wisde exactly when it equals P_i(n) for an appropriate integer n and some i.

Thus, infinitude of wisde primes is reduced to the unresolved problem of whether one of these irreducible cubics takes prime values infinitely often. The Bateman–Horn conjecture predicts an affirmative answer and an asymptotic count, but the paper does not establish it unconditionally.