We prove that for any partition $(\lambda_1,...,\lambda_{d^2})$ of size $dm$
there exists $k\ge 1$ such that the tensor square of the irreducible
representation of the symmetric group $S_{kdm}$ with respect to the rectangular
partition $(km,...,km)$ contains the irreducible representation corresponding
to the stretched partition $(k\lambda_1,...,k\lambda_{d^2})$. We also prove a
related approximate version of this statement in which the stretching factor
$k$ is effectively bounded in terms of $d$. This investigation is motivated by
questions of geometric complexity theory.