  We consider solutions of the massless scalar wave equation $\Box_g\psi=0$,
without symmetry, on fixed subextremal Kerr backgrounds $(\mathcal M, g)$. It
follows from previous analyses in the Kerr exterior that for solutions $\psi$
arising from sufficiently regular data on a two ended Cauchy hypersurface, the
solution and its derivatives decay suitably fast along the event horizon
$\mathcal H^+$. Using the derived decay rate, we show that $\psi$ is in fact
uniformly bounded, $|\psi|\leq C$, in the black hole interior up to and
including the bifurcate Cauchy horizon $\mathcal C\mathcal H^+$, to which
$\psi$ in fact extends continuously. In analogy to our previous paper, [30], on
boundedness of solutions to the massless scalar wave equation on fixed
subextremal Reissner--Nordstr\"om backgrounds, the analysis depends on weighted
energy estimates, commutation by angular momentum operators and application of
Sobolev embedding. In contrast to the Reissner--Nordstr\"om case the
commutation leads to additional error terms that have to be controlled.
