1-rotational Kirkman triple systems generate dicyclic Steiner 2-designs with block size 4

It is reasonable to conjecture that a 1-rotational ${\rm KTS}(2v+1)$ exists for any admissible $v$, i.e. for any $v\equiv1$ or $4\pmod{12}$. For the time being this is known to be true only for small $v$'s and for $v$'s whose prime factors are all $\equiv1\pmod{12}$. To prove the conjecture would be a valuable result in itself, but still more valuable considering that it would imply the existence of a regular $S(2,4,4v)$ for any $v\equiv1,4\pmod{12}$. In fact we prove that starting from any 1-rotational ${\rm KTS}(2v+1)$ it is possible to explicitly construct a regular $S(2,4,4v)$ over the dicyclic group.