The Fourier completeness criterion for affine contractive IFS measures

Let $\mu$ be an affine contractive IFS measure supported in $[0,1]^d$, and let $T=(T_{ij})$ be its corresponding Markov transition matrix. Let $v$ be a Perron–Frobenius vector satisfying

$$vT=v,\qquad \text{equivalently}\qquad \sum_j v_jT_{ji}=v_i.$$

Fourier completeness criterion. The Fourier frequencies $\{e_n\}_{n\in\mathbb{N}_0^d}$ are total in $L^2(\mu)$ if and only if $v$ is non-constant, namely, not proportional to $(1,1,\ldots,1)$.

This identifies completeness of the Fourier frequencies in the IFS $L^2$ space with a condition on the stationary Perron–Frobenius vector of the associated Markov transition matrix. The supplied text does not state whether the criterion is proved or remains open.