The main result in this paper is the proof of the recently conjectured non-existence of cubic Legendre multiplier sequences. We also give an alternative proof of the non-existence of linear Legendre multiplier sequences using a method that will allow for a more methodical treatment of sequences interpolated by higher degree polynomials. MSC 30C15, 26C10 Keywords: Legendre multiplier sequences, symbol of a linear operator, coefficients of Legendre-diagonal differential operators arXiv:1310.4563v2 [math.CV]