Generalized Mazya conjecture for weakly cancelling Fourier multipliers

Let $\mathrm{T}\colon S^{d-1}\to\mathbb{R}^{\ell}$ be sufficiently smooth, at least Hölder continuous, and weakly cancelling:

$$\int_{S^{d-1}}\mathrm{T}(\zeta)\,d\sigma(\zeta)=0.$$

For $\alpha\in(0,d)$, define

$$\mathbb{T}_\alpha[f]=\mathcal{F}^{-1}\left(\frac{\mathrm{T}(\xi/|\xi|)\widehat f(\xi)}{|\xi|^\alpha}\right),$$

let $\widetilde{\mathrm{T}}$ be determined by

$$\frac{\widetilde{\mathrm{T}}(x)}{|x|^{d-\alpha}}=\mathcal{F}^{-1}\left(\frac{\mathrm{T}(\xi/|\xi|)}{|\xi|^\alpha}\right)(x),$$

and choose $p$ so that $\frac{p-1}{p}=\frac{\alpha}{d}$. Let $\Phi\colon\mathbb{R}^{\ell}\to\mathbb{R}$ be positively $p$-homogeneous and locally Lipschitz. Generalized Mazya conjecture. The estimate

$$\left|\int_{\mathbb{R}^d}\Phi(\mathbb{T}_\alpha[f](x))\,dx\right|\lesssim\|f\|_{L_1}^p$$

holds for all compactly supported $f$ with $\int f=0$, with a uniform constant, if and only if

$$\int_{S^{d-1}}\Phi(\widetilde{\mathrm{T}}(x))\,d\sigma(x)=0.$$

The question generalizes Mazya's conjecture to fractional, vector-valued Fourier multipliers and is posed as an open question from the Bellman-function viewpoint; the source gives no resolution.