  We strengthen a conjecture by the author. This conjecture is a
Bogomolov-Gieseker type inequality involving the third Chern character of mixed
tilt-stable complexes on fibred threefolds. We extend it from complexes of
mixed tilt-slope zero to arbitrary relative tilt-slope. We show that this
stronger conjecture implies the support property of Bridgeland stability
conditions, and the existence of explicit stability conditions. We prove our
conjecture for ruled threefolds, hence improving a previous result by the
author.
