The Fourier-dimension conjecture for cones and cylinders generated by arbitrary sets

Let $A i\mathbb{R}^{d}$ be a set with Fourier dimension $\dim_F(A)=s\leq d$. Define the cone and cylinder generated by $A$ by

$$C_A=\{h(x,1)\mid x\in A, h\in\mathbb{R}\}\subset\mathbb{R}^{d+1},$$

and

$$D_A=\{(x,h)\mid x\in A, h\in\mathbb{R}\}\subset\mathbb{R}^{d+1}.$$

Cone-and-cylinder Fourier-dimension conjecture. Then

$$\dim_F(C_A)=\dim_F(D_A)=\dim_F(A)=s.$$

This question asks whether passing from an arbitrary set to the cone or cylinder it generates preserves Fourier dimension. The supplied text does not state whether the claim has been resolved.