I'm curious if there are any results that allow us to determine if a product of cyclotomic polynomials (not necessarily all distinct) results in a polynomial having nonnegative coefficients.

Some things that I do know: $\Phi_{p^k}(x)$ always has nonnegative coefficients for any prime $p$, and so any product of $\Phi_{p^k}(x)$'s will have positive coefficients. Whereas, if $m$ is not a power of a prime $\Phi_{m}(1) = 1$ despite having terms $x^{\phi(m)}$ and $1$ thus requiring at least one negative coefficient.

By computer I know some interesting examples like, $$\Phi_3(x)\Phi_6(x) = 1x^0 + 1x^2 + 1x^4$$ but $$\Phi_3(x)\Phi_6(x)^2 = 1x^0 + -1x^1 + 2x^2 + -1x^3 + 2x^4 + -1x^5 + 1x^6$$.

Adding a $\Phi_2(x)$ helps for a bit, $$\Phi_2(x)\Phi_3(x)\Phi_6(x) = 1x^0 + 1x^1 + 1x^2 + 1x^3 + 1x^4 + 1x^5$$ and $$\Phi_2(x)\Phi_3(x)\Phi_6(x)^2 = 1x^0 + 1x^2 + 1x^3 + 1x^4 + 1x^5 + 1x^7$$ both have nonnegative coefficients, but $$\Phi_2(x)\Phi_3(x)\Phi_6(x)^3 = 1x^0 + -1x^1 + 2x^2 + 1x^4 + 1x^5 + 2x^7 + -1x^8 + 1x^9$$ does not. However, I can't find any sort of rhyme or reason to the appearance of negative coefficients.

I've skimmed papers like http://math.ucsd.edu/~revans/PolynomialsGreene.pdf but can't find any sort of "if and only if" conditions. Being pointed even in a somewhat right direction would help me a lot.