  This manuscript deals with the three-dimensional version of a flux-limited
Keller-Segel system coupled to the incompressible Stokes equations through
transport and buoyancy.
  The main goal consists in verifying that within a certain parameter regime,
known as being optimal therefor in some fluid-free simplification, a feature of
blow-up prevention by suitably strong flux limitation persists also in the
framework of the considered full chemotaxis-fluid system. To achieve this, as a
secondary objective of possibly independent interest the manuscript separately
establishes some conditional bounds for fluid fields and taxis gradients in a
fairly general setting.
  The application thereof to the specific problem under consideration
thereafter facilitates the derivation of a result on global existence of
bounded classical solutions for widely arbitrary initial data actually, indeed
within the entire and essentially optimal parameter range, through an argument
which appears to be signficantly condensed when compared to reasonings pursued
in previous related works.
