Thought: (1) Since $0< f_n(x) \le n,$ I'm thinking if we can explore the topology of $y \in (0,n], f_n^{-1}(y)$. It seems to me that $\forall y \in (0,n], f_n^{-1}(y)$ consists of the disjoint union of at most $n$ points or hyperspheres $S^{p-1}\subset \mathbb{R}^p.$ whose supporting hyperplanes are orthogonal to the $x_n$ axis: so for example when $p=2, f_n^{-1}(y)$ is a disjoint union of at most $n$ points or horizontal circles whose supporting planes are orthogonal to the $x_3$ axis. If we can show that, then it'll show that for the local maximum $x*$ with local maximum value $y*, f_^{-1}(y*)$$y*, f_n^{-1}(y*)$ is a point or a disjoint union of points, as otherwise it'd be a disjoint union of hyperspheres orthogonal to the axis $x_{n+1}$ , which would mean that these hyperspheres can't come arbitrarily close to $x*,$ proving the local max is a strict one. Not sure if we need Morse theory to prove this?
Now can we say something about the above map $\Phi$ that guarantees that its fixed points are isolated? I don't know, but I'm looking at this and this question on MSE that deal with isolatedness and thus finitely many Lefscetz fixed points on a compact manifold, and in our case, we know that these fixed points of $\Phi: \mathbb{R}^p\to \mathbb{R}^p$ are on a compact manifold with boundary and corners, namely $C,$ the convex hull of $\{x_1, x_2 \dots x_n\}.$ So it seems to me that all we may need to show is that: $\Phi:C \to C$ is Lefschetz, i.e. $D\Phi_{x*}:\mathbb{R}^p\to \mathbb{R}^p$ does not have the eigenvalue $1.$ Can we show that, probably?
Thought (3) [after the suggestion in the first comment]
We can try proving the positive definiteness of the Hessian at the critical point $x*$,, where the Hessian $H(x)$ is as follows:
$$H(x)={\sum_{i=1}^{n}(-2I+4{(x-x_i)(x-x_i)^{T}})exp(-||{x-x_i}||^2)}.$$ But it seems difficult to manipulate at the critical point $x*$ above.
