Prill's problem

We solve Prill's problem, originally posed by David Prill in the late 1970s and popularized in ACGH's "Geometry of Algebraic Curves." That is, for any curve $Y$ of genus $2$, we produce a finite \'etale degree $36$ connected cover $f: X \to Y$ where, for every point $y \in Y$, $f^{-1}(y)$ moves in a pencil.