The followings are my recap of Musashi 2003 Taiga drama episodes. Spoilers for those who haven’t watched the (very great and totally recommended) series.
Soon after Musashi arrived in Kumamoto the Hosokawa father and son (Tadaoki vs Tadatoshi) organised the challenge between Musashi and Kojiro; both of them accepted the challenge.
Yagyu Munenori ordered to kill whoever wins in the duel between Musashi and Kojiro. So, basically, although Musashi and Kojiro had wanted to test their skills for a long time, their duel might have been different had it not been for Edo political feud.