Goncharov, Harizanov, Knight and Shore investigated the Turing degrees of $\Pi^1_1$ cofinal branches (which they call "paths through $\mathcal{O}$"). They showed there is a $\Pi^1_1$ cofinal branch which does not compute $\emptyset'$, so certainly doesn't compute true arithmetic. On the other hand, there is a $\Pi^1_1$ cofinal branch which computes $\mathcal{O}$.

Goncharov, Harizanov, Knight and Shore investigated the Turing degrees of $\Pi^1_1$ cofinal branches (which they call "paths through $\mathcal{O}$"). They showed there is a $\Pi^1_1$ cofinal branch which does not compute $\emptyset'$, so certainly doesn't compute true arithmetic. On the other hand, H. Friedman showed there is a $\Pi^1_1$ cofinal branch which computes $\mathcal{O}$ (reference can be found in the GHKS paper).