Question. Can there be an Aronszajn tree $T$, such that no c.c.c. forcing extension adds a cofinal branch to $T$, but there is an $\omega_1$-preserving forcing extension adding a cofinal branch to $T$?
To give some background, an Aronszajn tree is an $\omega_1$-tree with no $\omega_1$-branches, and it is special, if it is the union of countably many antichains. Any forcing extension in which a special Aronszajn tree $T$ comes to have a cofinal branch must collapse $\omega_1$, since the branch has size $\omega_1^V$ and will contain at most one node from each of the antichains witnessing that $T$ is special. In other words, a special Aronszajn tree must remain Aronszajn in any $\omega_1$-preserving forcing extension, and this includes all c.c.c. extensions. Meanwhile, any Aronszajn tree can be forced to be special, and the specializing forcing is c.c.c. and absolute (consisting of finite partial specializing functions). Thus, this forcing remains c.c.c. in any extension in which $T$ remains Aronszajn.
Concerning the question,
A special Aronszajn tree satisfies the first requirement of the question, since we cannot add a branch to it by c.c.c. forcing, but it doesn't satisfy the second requirement, since no $\omega_1$-preserving extension can add a cofinal branch. So special Aronszajn trees won't work.
A Souslin tree satisfies the second requirement, but not the first, precisely because it is itself c.c.c. and adds a branch through itself. So Souslin trees won't work.
What is needed is a tree that is almost special, in the sense that it remains Aronszajn in every c.c.c. extension, but not fully special, in the sense that we can add a cofinal branch by some (non-c.c.c.) $\omega_1$-preserving forcing.
I am interested in this question in part because the existence of such a tree $T$ would provide an answer Arthur Fischer's question on two versions of absolutely ccc, because the forcing $\mathbb{P}$ to specialize $T$ is c.c.c. and would remain c.c.c. in any c.c.c. extensions, since the tree would still be Aronszajn there, but it would not be c.c.c. in the $\omega_1$-preserving extension in which $T$ gains a cofinal branch.
