In his 1967 article "Sur un theoreme de R. Salem", Gatesoupe proved that if a set $A\subset [0,1]$ has Fourier dimension $\alpha$ then the set $\tilde A:=\{x\in \mathbb{R}^n: |x| \in A\}$ has Fourier dimension at least $n-1+\alpha$. Basically he starts from a measure $\mu$ on $A$ whose Fourier transform decays as $|\xi|^{-\alpha/2}$ and proves (via a rather standard argument with Bessel functions) that the measure $d\nu := r^{(1-n)/2} d\mu \otimes d\sigma^{n-1}$ has a Fourier transform that decays as $|\xi|^{-(n-1+\alpha)/2}$.

Is it known whether the equality between the two dimensions hold? i.e. $\dim_F(\tilde A) = n-1+\dim_F(A)$?

I would expect it to be the case but I could not prove it formally. I only have a partial result for some dimensions $n$, but I see no reason why this should be true e.g. for dimension $5$ and not dimension $3$ or $4$.