Suppose the pre-images of the two maxima are isolated this would be true and the proof involves using the Morse theory I think.
The idea is, define , then is nothing but , suppose are the two local maximal values , then with a small positive number , has at least two holes around and , for that , and since the critical points are isolated, by choosing sufficiently small, those holes will not connect each other to merge to a single hole. So the (n-1)th homology group of has at least dimension 2.
However since D is connected has only one hole inside, represented by a homology group of dimension 1.
According to Morse theory, between 0 and x, there must be other critical value z, so that the homotopic type of is got by gluing one or more (n-1) cells to , to increase the dimension of the (n-1)th homology group.
Then the theory says that the critical point z must have an index of (n-1), so it has an indefinite Hessian, with a signature of (1, -1, -1, ..., -1).