The Besov--Lorentz strengthening of the cancellation conjecture

With $\mathfrak{W}$, $\Omega$, $d$, $\ell$, $k$, and $S^{d-1}$ as above, for $\alpha\in(0,d)$, the Besov--Lorentz strengthening. The Riesz potential $\operatorname{I}_{\alpha}$ maps $\mathfrak{W}$ to $B^{0,1}_{d/(d-\alpha),1}$ if and only if

$$\bigcap_{\xi\in S^{d-1}}\Omega(\xi)=\{0\}.$$

This is proposed as a stronger form of the preceding Sobolev embedding conjecture, replacing the Lebesgue target by the sharper Besov--Lorentz space.