  The toral rank conjecture speculates that the sum of the Betti numbers of a
compact manifold admitting a free action of a torus of rank $r$ is bounded from
below by $2^r$. Clearly, such an action yields a torus bundle, and, more
generally, the same cohomological bound is conjectured for total spaces of
suitable topological torus fibrations by F\'elix--Oprea--Tanr\'e.
  In this article we show that this generalised toral rank conjecture cannot
hold by providing various different counter-examples to it (for each rank
$r\geq 5$). In particular, we show that there are sequences of smooth nilpotent
fibre bundles of nilmanifolds with fibre a torus of rank $r$ such that the
quotient of the total dimensions of the cohomologies of total space and fibre
even converges to $0$ with $r$ tending to infinity.
  We moreover prove that none of our torus fibrations can be realised by almost
free torus actions. More precisely, in the depicted sequence the difference of
the ranks of the torus fibres in the bundle and the toral ranks (which are
constant one) even tends to infinity.
  Similar to recent examples by Walker (of a completely different nature) this
shows that the toral rank conjecture is not likely to follow from "weaker
conjectures or structures".
