Suppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that \begin{equation*} \beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq C(1+|x|)^{-\gamma/2}\right\}. \end{equation*} The Fourier dimension is always majorized by the Hausdorff dimension (for sets, this is a consequence of Frostman's lemma) and various results in Geometric Measure Theory reveal that Fourier dimension is generally a much stronger and more stable property (for instance, when intersecting two sets, the Hausdorff "codimensions" need not add, but will if one of the sets has positive Fourier dimension, and the same holds in the context of dimensions of measures).
Because Fourier dimensions are generally much better behaved than Hausdorff dimensions, and it is immediate that the Hausdorff dimension does not drop when restricting to an open set, it is natural to ask whether this continues to hold true of Fourier dimension.
Is it true that for any measure $\mu$ of Fourier dimension $\beta$, and open set $U$, the Fourier dimension of $\mu_U$, the measure defined by \begin{equation*} \int g d\mu_U = \int_U g d\mu, \end{equation*} is also at least $\beta$?
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