  Let $T$ be a complete equicharacteristic local (Noetherian) UFD of dimension
$3$ or greater. Assuming that $|T| = |T/m|$, where $m$ is the maximal ideal of
$T$, we construct a local UFD $A$ whose completion is $T$ and whose formal
fibers at height one prime ideals have prescribed dimension between zero and
the dimension of the generic formal fiber. If, in addition, $T$ is regular and
has characteristic zero, we can construct $A$ to be excellent.
