Submatrix monotonicity of the Perron root

It is known that increasing an entry of a nonnegative matrix nondecreases (and generally increases) its Perron root. Motivated by a question raised by José Dias da Silva, we study the partial order on k-by-k nonnegative matrices in which A≾DSB if whenever A and B occur as submatrices in the same position in otherwise equal nonnegative matrices F and G, ρ(F)≤ρ(G). We find that this partial order is equivalent to the entry-wise partial order. This is proven with some asymptotic results about the Perron root that may be of independent interest.