Fractal-set Fourier-dimension conjecture for scalar products

Let $d$ be a positive integer, let $\beta\beta\in[0,d]$, and let $RY=\{ry:r\in R,\ y\in Y\}$ for $R\subseteq(0,\infty)$ and $Y\subseteq\mathbb{R}^d$. Fractal-set conjecture. For every $\beta\in[0,d]$, there exists a compact set $Y\subseteq\mathbb{R}^d$ such that

$$\dim_H(Y)=\dim_F(Y)=\beta$$

and, for every non-empty compact set $R\subseteq(0,\infty)$,

$$\dim_F(RY)\geq\min\{d,\dim_F(R)+\dim_F(Y)\}.$$

The unit sphere has this property by the preceding theorem, while arbitrary sets need not: sets contained in a hyperplane through the origin can have zero Fourier dimension after multiplication by $R$. The conjecture asks whether such sets exist at every prescribed Hausdorff and Fourier dimension.