In the following, I focus on trees of height $\omega_1$: if there exists a nonspecial tree any of whose $\aleph_1$-subtrees is special, must CH fail?

**Some neither consistent nor coherent thoughts**: Notice that the tree must have cardinality at least $\aleph_2$ and this is a fragment of $MA_{\aleph_1}$. It is known that it is possible to specialize Aronszajn trees with forcings that add no reals, hence obtaining a model of GCH along with all Aronszajn trees are special. But in this case, we are dealing with fat trees and maybe a possible attack is to think about forcings that specialize fat trees without adding reals.