Finiteness near the corrected Beiter bound

For each prime $p$, let

$$M(p):=\max_{p<q<r\text{ primes}} A(pqr),$$

\nwhere $A(pqr)$ is the height of $\Phi_{pqr}$.

Finiteness conjecture. For any real number $r$, there exist only finitely many primes $p$ such that

$$M(p)>\frac{2p}{3}-r.$$

This is a stronger formulation of the proposed unbounded sharpening of the corrected Beiter bound. The conjecture remains open in the supplied text.