The height-realisation conjecture for ternary cyclotomic polynomials

For an odd prime $p$, define

$$M(p)\coloneqq\max_{q,r} A(pqr).$$

Height-realisation conjecture. For every odd prime $p$ and every integer $h$ with $1\le h\le M(p)$, there exist primes $q$ and $r$ such that

$$A(pqr)=h.$$

The conjecture asserts that, for each fixed odd prime $p$, every height up to the maximum height $M(p)$ is attained by a ternary cyclotomic polynomial of the form $\Phi_{pqr}$. The preceding results show that every positive integer occurs as the height of some $\Phi_{pqr}$, but they do not establish this stronger fixed-$p$ realisation statement.