On subtensors of high partition rank

We prove that for every positive integer $d \ge 2$ there exist polynomial functions $F_d, G_d: \mathbb{N} \to \mathbb{N}$ such that for each positive integer $r$, every order-$d$ tensor $T$ over an arbitrary field and with partition rank at least $G_d(r)$ contains a $F_d(r) \times \cdots \times F_d(r)$ subtensor with partition rank at least $r$. We then deduce analogous results on the Schmidt rank of polynomials in zero or high characteristic.