Real eigenvectors of regular simplex tensors

Let $\mathcal{T}$ be a regular simplex tensor generated by a regular simplex frame. An eigenvector is a nonzero vector $\mathbf{x}$ satisfying

$$\mathcal{T}(\mathbf{x},\ldots,\mathbf{x},\cdot)=\lambda\mathbf{x}$$

for some scalar $\lambda$, with eigenvectors considered up to scaling. The real-eigenvector conjecture. All eigenvectors of a regular simplex tensor are complex, except for the vectors in the frame and, when $d\geq 6$ is even, the vectors on the boundary of the regions of convergence. The conjecture concerns the complete eigenvector lists obtained from the polynomial eigenvector equations; the source presents it as suggested by the computed tables and does not report a proof or resolution.