  Let $H$ be a finite-dimensional weak Hopf algebra over a field $k$ and $A/B$
be a right faithfully flat weak $H$-Galois extension. We prove that if the
finitistic dimension of $B$ is finite, then it is less than or equal to that of
$A$. Moreover, suppose that $H$ is semisimple. If the finitistic dimension
conjecture holds, then the finitistic dimension of $B$ is equal to that of $A$.
