Combined Maximality Principles up to large cardinals

The motivation for this paper is the following: In [Fuc08] I showed that it is inconsistent with ZFC that the maximality principle for closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the maximality principle for <κ-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of∞), and if so, how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high. As a by-product, assuming the consistency of a supercompact cardinal, I show that it is consistent that the least weakly compact cardinal is indestructible.