Is there a limit computable function $\Phi$ with the following properties?
Let $T\subset \mathbb{N}^{<\mathbb{N}}$ be a tree (coded as its characteristic function) and $f\in\mathbb{N}^\mathbb{N}$ be strictly increasing. Let also $T_f$ be the subtree of $T$ whose strings are dominated by $f$, i.e. $$ \sigma \in T_f \iff \sigma \in T \text{ and } (\forall i<|\sigma|)(\sigma(i)\le f(i)) $$ For every $T$ and $f$ as above $\Phi^{T\oplus f}$ produces a path through $T_f$ if there is one, otherwise produces any infinite string.
For every $T$ and every $f$ as above there is a substring $g$ of $f$ s.t. $\Phi^{T\oplus g}$ requires at most $g(i)$ steps to stabilize on the $i$-th component.
Notice that if $f$ is strictly increasing then every substring $g$ of $f$ dominates $f$.
The first condition is easy to satisfy, the second one is harder and I fear there is no $\Phi$ that satisfies both. I'm thinking of $\Phi$ as a computable map that produces a sequence $(p_i)_{i\in\mathbb{N}}\subset\mathbb{N}^\mathbb{N}$ converging to the solution (equivalently you can think of $\Phi$ as a machine that is allowed to change its output finitely many times for each component).
