On the computational power of $C$-random strings

Denote by $H$ the Halting problem. Let $R_U: = \{ x | C_U(x) \ge |x|\}$, where $C_U(x)$ is the plain Kolmogorov complexity of $x$ under a universal decompressor $U$. We prove that there exists a universal $U$ such that $H \in P^{R_U}$, solving the problem posted by Eric Allender.