The small-ball to Fourier-decay conjecture for polynomial images

Let $X\sim\mu$, where $\mu$ is a log-concave measure on $\mathbb{R}^n$, and let $f:\mathbb{R}^n\to\mathbb{R}$ be a polynomial of degree $d$. For a Borel measure $\nu$ on $\mathbb{R}$, write $\mathcal{F}(\nu)(t)=\int e^{\mathrm{i}tx}\,d\nu(x)$, and let $f_*\mu$ denote the pushforward of $\mu$ under $f$. The small-ball to Fourier-decay conjecture. Suppose that for some $C>0$ and some $\alpha\in(1/d,1)$,

$$\sup_{a\in\mathbb{R}}\mathbb{P}\left(|f(X)-a|\leq\varepsilon\right)\leq C\varepsilon^\alpha,\qquad\text{for every }\varepsilon>0.$$

Then there is a constant $C'$, depending only on $C$ and $d$, such that

$$|\mathcal{F}(f_*\mu)(t)|\leq\frac{C'}{|t|^\alpha},\qquad\text{for every }t\neq0.$$

This would improve the general exponent $1/d$ in the Fourier-decay bound for polynomial images when stronger small-ball estimates are available. The surrounding discussion presents it as an open question, and no resolution is supplied.