Conjectural nonnegative slopes for locally analytic overconvergent cohomology

Fix $w\in{}^MW$, and let $\nu:T^c(\mathbb{Z}_p)\to\mathbb{C}_p^\times$ be a continuous character. For a character $\lambda$ of $T^+$ or $T^-$, let $v(\lambda)$ denote its slope in the relevant real character space. Slope conjecture. Every character $\lambda$ of $T^+$ occurring on $\mathrm{R}\Gamma_{w,\mathrm{an}}(K^p,\nu)^{+,fs}$ or $\mathrm{R}\Gamma_{w,\mathrm{an}}(K^p,\nu,cusp)^{+,fs}$ satisfies

$$v(\lambda)\geq 0.$$

Every character $\lambda$ of $T^-$ occurring on $\mathrm{R}\Gamma_{w,\mathrm{an}}(K^p,\nu)^{-,fs}$ or $\mathrm{R}\Gamma_{w,\mathrm{an}}(K^p,\nu,cusp)^{-,fs}$ satisfies

$$v(\lambda)\leq 0.$$

The conjecture concerns slope bounds for locally analytic overconvergent cohomology and is stated to be consistent with the paper's stronger slope conjecture; the supplied text gives no resolution status.