Orders of elements and zeros and heights of characters in a finite group

Let \chi be an irreducible character of the finite group G. If g is an element of G and \chi(g) is not zero, then we conjecture that the order of g divides |G|/\chi(1). The conjecture is a generalization of the classical fact that irreducible p-projective characters vanish on p-singular elements, since the latter is equivalent to saying that if \chi(g) is not zero then the square free part of the order of g divides |G|/\chi(1). We prove some partial results on the conjecture; in particular, we show that the order of g divides (|G|/\chi(1))^2. Using these results, we derive some bounds on heights of characters. We also pose a related conjecture concerning congruences satisfied by central character values.