Real spectral-radius eigenvalue conjecture for ALS on canonical tensor decomposition

Let $q$ be the ALS fixed-point map for canonical tensor decomposition, and let $x^*$ be a fixed point. Write $q'$ for the Jacobian evaluated at $x^*$, and let $\rho_{q'}$ denote its spectral radius after excluding the eigenvalues equal to $1$ caused by Hessian degeneracy at $x^*$. Real spectral-radius eigenvalue conjecture. There exists a real eigenvalue $\mu$ of $q'$ such that

$$\rho_{q'}=\mu.$$

This conjecture is motivated by numerical tests for ALS on canonical tensor decomposition; the cited work indicates that the relevant spectral-radius condition is known for ALS, while its general validity in this setting remains a topic for further research.