On the order of the best approximation in spaces with asymmetric norm and sign-sensitive weight on classes of differentiable functions

The class of asymmetric norms with sign-sensitive weight contains both the classical norms of the spaces $L_p(\mathbb T)$ and the metrics that generate one-sided approximations. For sign-sensitive weights $\varrho$, $\tilde\varrho$ and an asymmetric monotone norm $\psi(u,v)$ on the plane, we obtain an upper estimate for the number
$$ E_n(\mathrm{B}\mathrm{W}_{\psi_{\boldsymbol{\varrho},\mathbf{p}}}^r(\mathbb T),L_{\psi_{\tilde{\boldsymbol{\varrho}},\mathbf{q}}}(\mathbb T))=\sup_{f\in\mathrm{B}\mathrm{W}_{\psi_{\boldsymbol{\varrho},\mathbf{p}}}^r(\mathbb T)}\inf_{t\in T_n}\psi_{\tilde{\boldsymbol{\varrho}},\mathbf{q}}(f(\,\cdot\,)-t(\,\cdot\,)). $$
In some important cases of asymmetric norms with fixed sign-sensitive weights $\varrho=(\alpha,\beta)$, we find the rate of decrease of this number as $n\to+\infty$ for a fixed $r\in\mathbb N$.