Carbery–Wright oscillatory integral conjecture

Let $n,d\in\mathbb{N}$, let $\mathcal{P}_d(\mathbb{R}^n)$ denote the polynomials in $n$ variables of degree at most $d$, and let $f\in\mathcal{P}_d(\mathbb{R}^n)$ satisfy

$$\int_{[0,1]^n}f(x)\,dx=0,\qquad \int_{[0,1]^n}|f(x)|\,dx=1.$$

Carbery–Wright conjecture. There exists an absolute constant $C>0$ such that

$$\left|\int_{[0,1]^n}e^{itf(x)}\,dx\right|\leq \frac{C\min\{d,n\}}{|t|^{1/d}}\quad\forall t\in\mathbb{R}\setminus\{0\}.$$

This conjecture is an oscillatory-integral formulation motivated by sharp polynomial sublevel-set estimates on convex bodies. The supplied source does not state whether it has been resolved.