Is it true that if $\kappa < \kappa^\nu$, then ${\mathrm{cf}(\kappa)} \leq \nu$?

Question

Let $\kappa$ denote an infinite cardinal number. Then we know the following. $$\kappa<\kappa^{\mathrm{cf}(\kappa)}$$

Question. Is it true that if $$\kappa < \kappa^\nu,$$ then ${\mathrm{cf}(\kappa)} \leq \nu$?

Answer 1

No, it's not true at all.

Consider the case $2^{\aleph_0}=\aleph_2$ and take $\kappa=\aleph_1$ and $\nu=\aleph_0$.