Disjoint base-block conjecture for cyclic $(v,k,1)$-designs

Let $(V,\mathcal{B})$ be a cyclic $(v,k,1)$-design: the blocks are partitioned into orbits under a cyclic automorphism, and a base block is one chosen block from each orbit. Disjoint base-block conjecture. For every cyclic $(v,k,1)$-design, one can choose one block from each block orbit so that all chosen blocks are pairwise disjoint. This extends the Steiner triple system case to arbitrary block size $k$ and allows short orbits; the paper proves the assertion when $v$ is prime, but leaves the general case conjectural.