Remember a degree $\mathbb{d}$ is the $n$-lub of $\mathbb{c}_j$ in the Turing degrees if it is the least element (not merely a minimal element) set of $\mathbb{c}^{(n)}$ such that $\mathbb{c}$ computes every $\mathbb{c}_j$. It is non-trivial if it's not the $n$-th jump of a finite join of the degrees $\mathbb{c}_n$.

Enderton, Putnam and Sacks have proved $0^{\omega}$ is the (non-trivial) 2-lub of $0^n$. Are there any arithmetic degrees that are non-trivial 2-lub?

If so can they be arithmetic? What about an upper bound $\mathbb{c}$ of $\mathbb{c}_j$ with $\mathbb{c}$ of r.e. degree and $\mathbb{c}^{(2)}$ a non-trivial 2-lub of $\mathbb{c}_n$?