N-spherical functors and tensor categories

We apply the recently introduced notion, due to Dyckerhoff, Kapranov and Schechtman, of $N$-spherical functors of stable infinity categories, which generalise spherical functors, to the setting of monoidal categories. We call an object $N$-bounded if the corresponding regular endofunctor on the derived category is $N$-spherical. Besides giving new examples of $N$-spherical functors, the notion of $N$-bounded objects gives surprising connections with Jones-Wenzl idempotents, Frobenius-Perron dimensions and central conjectures in the field of symmetric tensor categories in positive characteristic.