Sun's conjecture on transversals in 3-dimensional latin hypercubes

A latin hypercube of dimension $d$ and order $n$ is the $d$-dimensional analogue of a latin square, with every symbol occurring exactly once on each axis-parallel line. Sun's conjecture. Every $3$-dimensional latin hypercube has a transversal. Here a transversal is a set of $n$ entries meeting every coordinate hyperplane and containing every symbol exactly once. The conjecture is equivalent to positivity of the permanent for the corresponding $4$-dimensional polystochastic $(0,1)$-matrices; it remains open.