The dimension formula for multi-scale constructions

Let $A(y)$ be the almost periodic filter represented by a polynomial $P$, let $M(A)=M(P)$ be its Mahler measure, and let $\lambda$ be the scaling parameter. For a Borel measure $f$, define

$$\rho(f)=-\frac{\ln M(A)}{\ln\lambda}.$$

Its Hausdorff dimension is denoted by $\operatorname{dim}(f)$. The dimension formula. If $f$ is a Borel measure, then

$$\rho(f)=\operatorname{dim}(f).$$

The preceding result identifies $\rho(f)$ with the asymptotic logarithmic growth rate of the Fourier transform, so this claim would connect the Mahler measure of the filter to the Hausdorff dimension of the associated measure. The supplied text does not state whether the formula is proved or remains open.