  We prove the Strominger-Thompson quantum Bousso bound in the infinite class
of conformal vacua in semiclassical JT gravity, with postive or negative
cosmological constant. The Bousso-Fisher-Leichenauer-Wall quantum Bousso bound
follows from an analogous derivation, requiring only initial quantum
non-expansion. In this process, we show that the quantity ${2\pi
k^{\mu}k^{\nu}<:T_{\mu\nu}:>-S''-\frac{6}{c}(S')^2}$ vanishes in any vacuum
state, entailing a stronger version of Wall's quantum null energy condition. We
derive an entropy formula in the presence of a generic class of two reflecting
boundaries, in order to apply our argument to the half reduction model of de
Sitter JT gravity.
