Enumeration of Nondegenerate $2\times (k+1)\times k$ Hypermatrices

We consider the problem of enumerating hypermatrices of format $2\times (k+1)\times k$ over a finite field that have nonzero hyperdeterminant and whose nonzero entries are restricted to a plane partition. We conjecture an attractive product formula for the enumeration, and prove it in many cases. In general, we show that the enumeration is given (up to a power of $q-1$) by a polynomial in $q$ with nonnegative integer coefficients, whose value at $q=1$ enumerates a natural family of three-dimensional rook placements.