Conjectural slope bounds for finite-slope cohomology

Let $w\in{}^MW$, $\kappa\in X^\star(T^c)^{M_\mu,+}$, and let $\chi:T(\mathbb{Z}_p)\to {\overline{F}}^\times$ be a finite-order character. For a character $\lambda$ of $T^+$ or $T^-$, let $v(\lambda)$ denote its slope, viewed in $X^\star(T^d)_\mathbb{R}$, and let $\rho$, $w_{0,M}$, and the Weyl-group action be as in the setup. Slope-bound conjecture. For any character $\lambda$ of $T^+$ occurring on $\mathrm{R}\Gamma_w(K^p, \kappa,\chi)^{+,fs}$ or $\mathrm{R}\Gamma_w(K^p, \kappa,\chi,cusp)^{+,fs}$, one has

$$v(\lambda) \geq w^{-1}w_{0,M}(\kappa+\rho)+\rho.$$

For any character $\lambda$ of $T^-$ occurring on $\mathrm{R}\Gamma_w(K^p, \kappa,\chi)^{-,fs}$ or $\mathrm{R}\Gamma_w(K^p, \kappa,\chi,cusp)^{-,fs}$, one has

$$v(\lambda) \leq w^{-1}(\kappa+\rho)-\rho.$$

These bounds are proposed as general lower and upper slope estimates for the finite-slope cohomology and its cuspidal part; the supplied text does not indicate whether they have been proved or disproved.