A polyhedron $P$ in $3$-space is said to be $1$-fillable in direction $-z$ provided that for every point of $P$ there is a $+z$-monotone path from it to the $z$-maximum of $P$. Thus, a polyhedron is $1$-fillable if there is an orientation of $P$ in which it is $1$-fillable. Given a simple polyhedron $P$, decide whether or not $P$ is $1$-fillable.