  We formulate conjectures regarding the maximum value and maximizing matrices
of the permanent and of diagonal products on the set of stochastic matrices
with bounded rank. We formulate equivalent conjectures on upper bounds for
these functions for nonnegative matrices based on their rank, row sums and
column sums. In particular we conjecture that the permanent of a singular
nonnegative matrix is bounded by 1/2 times the minimum of the product of its
row sums and the product of its column sums, and that the product of the
elements of any diagonal of a singular nonnegative matrix is bounded by 1/4
times the minimum of the product of its row sums and the product of its column
sums.
