Existence of pointclasses realizing intermediate least unreachable cardinals

Let $\lambda$ be a cardinal with $\bf{\delta^1_4}\leq\lambda\leq\bf{\delta^1_6}$, and suppose $\operatorname{cof}(\lambda)>\omega$. A cardinal is $\bf{\Gamma}$-unreachable when there is no sequence of distinct $\bf{\Gamma}$-sets of length that cardinal. Intermediate-unreachability conjecture. There is a pointclass $\bf{\Gamma}$ such that $\lambda$ is the least $\bf{\Gamma}$-unreachable cardinal. The source presents this as an open question about unreachability, although it also notes a theorem asserting the corresponding conclusion in the indicated interval under the stated cofinality condition.