Ternary cyclotomic polynomials with an optimally large set of coefficients

Ternary cyclotomic polynomials are polynomials of the form $\Phi_{pqr}(z)=\prod_\rho(z-\rho)$, where $p<q<r$ are odd primes and the product is taken over all primitive $pqr$-th roots of unity $\rho$. We show that for every $p$ there exists an infinite family of polynomials $\Phi_{pqr}$ such that the set of coefficients of each of these polynomials coincides with the set of integers in the interval $[-(p-1)/2,(p+1)/2]$. It is known that no larger range is possible even if gaps in the range are permitted.