Tkachenko's conjecture on weakly pseudocompact precompact groups

Let $G$ be a subgroup of a compact group. A space is weakly pseudocompact if it is $G_\delta$-dense in some compact space, meaning that every non-empty $G_\delta$-subset of that compact space intersects it; a space is pseudocompact if and only if it is $G_\delta$-dense in its Stone–Čech compactification $\beta G$. Tkachenko's conjecture. If $G$ is $G_\delta$-dense in some compact space, then $G$ is $G_\delta$-dense also in its Stone–Čech compactification $\beta G$. Equivalently, every weakly pseudocompact precompact topological group is pseudocompact. The conjecture asks whether weak pseudocompactness and pseudocompactness coincide for precompact topological groups; the supplied source does not state a resolution.