High-rank subtensors of high-rank tensors

Let $d \ge 2$ be an integer. We show that for a class of notions $R$ of rank for order-$d$ tensors, which includes in particular the tensor rank, the slice rank and the partition rank, there exist functions $F_{d,R}$ and $G_{d,R}$ such that if an order-$d$ tensor has $R$-rank at least $G_{d,R}(l)$ t