Generic projection invariance of Euclidean distance degree for arbitrary varieties

Let $X\subseteq\mathbb{R}^n$ be an affine variety, let $K\subseteq\mathbb{R}^n$ be a generic linear subspace with $\dim K=k<n-d$, where $d=\dim X$, and let $Y=\operatorname{cl}\pi(X)$, with $\pi:\mathbb{R}^n\to K^\perp$ the orthogonal projection. Generic projection invariance conjecture. Fact~ holds for arbitrary varieties; equivalently, the generic Euclidean distance degrees satisfy

$$\operatorname{gEDD}(Y)=\operatorname{gEDD}(X).$$

The stated fact is known when $X$ is an affine cone over a projective variety, but the authors are not aware of a proof for arbitrary varieties; establishing this would contribute fundamentally to the theory of Euclidean distance degrees.