The Fourier-decay conjecture for non-degenerate analytic images of self-similar measures

Let $k,d$ be positive integers. Let $\mu$ be a self-similar measure on $\mathbb{R}^k$ which is not supported in any proper affine hyperplane. Let $U\subset\mathbb{R}^k$ be an open neighbourhood of $\operatorname{supp}(\mu)$, and let $f\colon U\to\mathbb{R}^d$ be real-analytic. The map $f$ is non-degenerate when the graph of $f$ is not contained in any proper affine hyperplane of $\mathbb{R}^{k+d}$.

Fourier-decay conjecture. There exists $\sigma>0$ such that

$$|\widehat{\mu_f}(\boldsymbol{\xi})|\ll |\boldsymbol{\xi}|^{-\sigma}.$$

The conjecture would answer the preceding questions by removing the stronger hypotheses used for the quantitative theorem, including the condition $\kappa_2>1$ and holomorphicity. It predicts polynomial Fourier decay for all such non-degenerate real-analytic pushforwards.