Feng–Horsley–Wang conjecture on disjoint base blocks in cyclic BIBDs

Let $k$ and $\lambda$ be fixed positive integers with $k\geqslant \lambda+1$. Let $(X,\mathcal B)$ be a cyclic $(v,k,\lambda)$-BIBD with $v\geqslant v_0$, where $v_0$ is an integer depending on $k$ and $\lambda$. Partition its blocks into orbits under the cyclic automorphism, and call a fixed block from each orbit a base block. Feng–Horsley–Wang conjecture. It is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. The source records a theorem resolving the assertion when $k\geqslant 2\lambda+1$, while the boundary range $\lambda+1\leqslant k\leqslant 2\lambda$ is not resolved there.