It seems to be one of those things that just seem unsaid but obvious in The Constitution. If there was no obligation to try it, the majority leader would hold absolute power.
Yet, the Senate has,, in the past, used committees to gather evidence and question witnesses, rather than holding what we normally would consider a "trial". 2 judges who were impeached appealed their verdicts on those grounds. SCOTUS ruled that the the Senate could conduct "trials" any way they wanted to.
So, no, they are not bound to hold a "trial" in the conventional sense.