Conjecture on robust eigenvectors of regular simplex tensors

Let $\mathbf{w}_1,\ldots,\mathbf{w}_{n+1}\in\mathbb{R}^n$ be the vectors of a regular simplex frame, and let

$$\mathcal{S}:=\sum_{i=1}^{n+1}\mathbf{w}_i^{\circ m}$$

be the associated symmetric tensor of order $m$ and dimension $n$, where $\circ$ denotes the outer product. A robust eigenvector is an eigenvector of the tensor power method whose fixed point is robust under the relevant perturbations. Robust-eigenvector conjecture. The robust eigenvectors of a regular simplex tensor are precisely the vectors in the frame. The paper studies this claim for regular simplex tensors; the supplied text does not state whether it has been resolved.