The Sobolev gradient convolution trace inequality conjecture

Let $K\colon \mathbb{R}^d\to\mathbb{R}^d$ be homogeneous of order $1-d$ and smooth outside the origin. Let $\nu$ be a measure satisfying

$$\nu(B_r(x))\lesssim r^{d-1}.$$

For $f\in C_0^\infty(\mathbb{R}^d)$, consider the convolution $K*\nabla f$. Sobolev gradient convolution trace inequality conjecture. The inequality

$$\|K*\nabla f\|_{L_1(\nu)}\lesssim \|\nabla f\|_{L_1(\mathbb{R}^d)}$$

holds for all such measures $\nu$ if and only if, for every $e\in S^{d-1}$,

$$\int_{e^\perp\cap S^{d-1}}\langle K(\zeta),e\rangle\,d\zeta=0.$$

Here the integral uses the natural Hausdorff measure on the $(d-2)$-dimensional sphere $e^\perp\cap S^{d-1}$. The conjecture is motivated by the finite-dimensional cancellation theorem for gradient operators, but the source does not state a resolution.