  In 2009, Etzion and Siberstein proposed a conjecture on the largest dimension
of a linear space of matrices over a finite field in which all nonzero matrices
are supported on a Ferrers diagram and have rank bounded below by a given
integer. Although several cases of the conjecture have been established in the
past decade, proving or disproving it remains to date a wide open problem. In
this paper, we take a new look at the Etzion-Siberstein Conjecture,
investigating its connection with rook theory. Our results show that the
combinatorics behind this open problem is closely linked to the theory of
$q$-rook polynomials associated with Ferrers diagrams, as defined by Garsia and
Remmel. In passing, we give a closed formula for the trailing degree of the
$q$-rook polynomial associated with a Ferrers diagram in terms of the
cardinalities of its diagonals. The combinatorial approach taken in this paper
allows us to establish some new instances of the Etzion-Silberstein Conjecture
using a non-constructive argument. We also solve the asymptotic version of the
conjecture over large finite fields, answering a current open question.
