A symmetric positive semi-definite (PSD) tensor, which is not sum-of-squares (SOS), is called a PSD non-SOS (PNS) tensor. Is there a fourth order four dimensional PNS Hankel tensor? The answer for this question has both theoretical and practical significance. Under the assumptions that the generating vector of a Hankel tensor is symmetric and the fifth element of is fixed at , we show that there are two surfaces and with the elements of as variables, such that , is SOS if and only if , and is PSD if and only if , where is the first element of . If for a point , there are no fourth order four dimensional PNS Hankel tensors with symmetric generating vectors for such . Then, we call such a PNS-free point. We prove that a -degree planar closed convex cone, a segment, a ray and an additional point are PNS-free. Numerical tests check various grid points and report that they are all PNS-free.