I am wondering if a normal $\omega_1$-Aronszajn tree that has a big antichain (of size $\omega_1$) $A$ with the set $$ \mathrm{rk} (A) := \{ \theta: \exists x \in A ( \mathrm{ht} (x) = \theta )\}$$ club in $\omega_1$ is consistent with ZFC.
To my knowledge it is consistent that $\mathrm{rk} (A)$ be stationary. I was trying to produce a branch under the assumptions but got stuck. Any thoughts are appreciated.