Endpoint trace inequality characterization for divergence-free charges

Let $F$ be an $\mathbb{R}^d$-valued charge with $\operatorname*{\div}F=0$, let $u$ be a measure on $\mathbb{R}^d$, and let $K_1,K_2,\ldots,K_d$ be sufficiently smooth functions on $\mathbb{R}^d$ homogeneous of order $-1$. Define

$$K[F]=\sum_{j=1}^d K_j*F_j.$$

The endpoint trace inequality

$$\int_{\mathbb{R}^d}|K[F]|\,d\nu\lesssim |F|(\mathbb{R}^d)\|\nu\|_{\mathcal{M}^{1}(\mathbb{R}^d)}$$

holds if and only if

$$\sum_{j=1}^d K_j(\xi)\xi_j=0$$

for every $\xi\in\mathbb{R}^d$. This proposes a characterization of the cancellation condition needed for the endpoint estimate, whose validity remains unclear in the general setting.