The interplay between bounded ranks of tensors arising from partitions

Let $$ be integers. Using a fragmentation technique, we characterise $$-tuples $$ of non-empty families of partitions of $$ such that it suffices that an order-$$ tensor has bounded $$-rank for each $$ for it to have bounded $$-rank. On the way, we prove power lower bounds on suitable products of diagonal tensors, providing a qualitative answer to a question of Naslund.