Tangent-cone cancellation and Maz'ya's $Φ$-inequalities on finitely cornered planar domains

Let $0<α<2$, $p=2/(2-α)$, and let $K:\mathbb{R}^2\setminus\{0\}\to\mathbb{R}^m$ and $Φ:\mathbb{R}^m\to\mathbb{R}$ be positively homogeneous of degrees $α-2$ and $p$, with Lipschitz angular parts. For bounded finitely cornered piecewise-$C^{1,β}$ planar domains $Ω$, we characterize the critical estim