On the number of terms in the irreducible factors of a rational polynomial

For an integer m > 3, does there exist an absolute constant K(m) such that every polynomial with m non-zero coefficients has an irreducible factor with at most K(m) coefficients? A previous result in the literature establishes K(3) > 9, which is here improved to K(3) > 12. Improvements on known bounds are also given for m = 4, 5, 6, and for K(m), when m > 7.