Recall Silver's theorem which says that "if GCH holds below $\aleph_{\omega_1}$, then $2^{\aleph_{\omega_1}}=\aleph_{\omega_1+1},$ i.e., it also holds at $\aleph_{\omega_1}$".

Recently, Gitik has proved a similar result for collapses. See his paper Silver type theorems for collapses.

Here I would like to ask two related questions:

Question 1. Suppose that for each limit ordinal $\alpha < \omega_1,$ there exists a special $\aleph_{\alpha+1}$-Aronszajn tree. Does it follow that there is a special $\aleph_{\omega_1+1}$-Aronszajn tree?

The next question is related to tree property.

Question 2. Suppose that for each limit ordinal $\alpha < \omega_1,$ there exists an $\aleph_{\alpha+1}$-Aronszajn tree. Does it follow that there is an $\aleph_{\omega_1+1}$-Aronszajn tree?