Question: Does every $\omega_1$-Aronszajn tree contains a Suslin sub-tree or a special Aronszajn sub-tree?
Recall that Suslin trees are $\omega_1$-trees (trees of height $\omega_1$, and countable width) that have only countable anti-chains. Special Aronszajn trees are $\omega_1$-trees that can be represented as a countable union of anti-chains. Aronszajn trees are $\omega_1$-trees without branches.
Every sub-tree of a special Aronszajn tree with height $\omega_1$ is special, and therefore - not Suslin. Similarly, every subtree of a Suslin tree is Suslin so it's not special.
So, my question is if those are the only possibilities or that (consistently) there are other kinds of Aronszajn trees in which every sub-tree is not a Suslin tree and not a special Aronszajn tree.
Note that under $MA$, for example, every Aronszajn tree is special, so it is consistent that the answer is (trivially) positive.
Edit: In this question, a sub-tree is a subset with the restricted order, namely if $\langle T,<_T\rangle$ is a tree, then for every $X\subset T$, $\langle X, <_T \cap (X\times X)\rangle$ is a subtree (usually not an $\omega_1$-tree by itself).
Sub Question: Is it consistent that there is an $\omega_1$-tree that has no special sub tree but it is not Suslin?
Mohammad Golshani asked this question in the comments. Maybe a solution for the sub-question will lead to an answer for the main question.