Negation of CH implied by lots of special subtrees?

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.

Rado's conjecture is a combinatorial statement about instances of compactness in chromatic numbers of certain graphs, but Todorčević proved that it is equivalent to the statement that any tree, all of whose subsets of size $\aleph_1$ are special, is itself special.
So a counterexample to Rado's conjecture is a nonspecial tree, all of whose subsets of size $\aleph_1$ are special. In particular, all of its $\aleph_1$-subtrees are special.
In the same paper Todorčević also proves that Rado's conjecture implies that for every regular $\kappa>\omega_1$, every stationary subset of $\kappa\cap \operatorname{Cof}(\omega)$ reflects. Since the usual forcing to add a nonreflecting stationary set does not add reals, we can start in a model of CH and force the failure of Rado's conjecture, and consequently a tree as in your question, while preserving CH.