Positivity and definiteness of dominant transfer-matrix eigenvectors for real iTRs

Let $\bfx$ be a real translationally invariant infinite tensor ring (iTR) with core slices $X(i)\in\mathbb{R}^{r\times r}$, and let

$$T_X=\sum_{i=1}^d X(i)\otimes X(i).$$

Assume that the dominant eigenvalue of $T_X$ is simple, and write the corresponding left and right eigenvectors as $v_L=\operatorname{vec}(V_L)$ and $v_R=\operatorname{vec}(V_R)$.

Positivity and definiteness conjecture. The dominant eigenvalue $\eta$ is real and positive, and

$$\sum_{i=1}^d X(i)^{\mathsf T}V_LX(i)=\eta V_L,\qquad \sum_{i=1}^d X(i)V_RX(i)^{\mathsf T}=\eta V_R,$$

where $V_L$ and $V_R$ are positive definite matrices.

This claim describes the Perron-type structure expected for the transfer matrix governing operations on translationally invariant infinite tensor rings. The supplied parser status is unknown, so the resolution of the claim should be checked in the source or subsequent literature.