Dual fractal-set Fourier-dimension conjecture for scalar products

Let $d$ be a positive integer, let $RY=\{ry:r\in R,\ y\in Y\}$ for $R\subseteq(0,\infty)$ and $Y\subseteq\mathbb{R}^d$. Dual fractal-set conjecture. For every $\alpha\in[0,1]$, there exists a compact set $R\subseteq(0,\infty)$ such that

$$\dim_H(R)=\dim_F(R)=\alpha$$

and, for every compact set $Y\subseteq\mathbb{R}^d$ containing a non-zero point,

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

This is presented as the dual form of the preceding conjecture. It seeks a single scalar set $R$ of each prescribed dimension that gives the expected lower bound uniformly for all compact sets $Y$ containing a non-zero point.