This is answered affirmatively by Odlyzko and Richmond, *On the unimodality of high convolutions of discrete distributions*, Annals of probability (1985) 299--306:  all sufficiently large powers of the polynomial (with positive coefficients and no gaps) are strongly unimodal, that is, the coefficients form a log convex sequence. The proof uses estimates of contour integrals over circles of just the right radius.