Hemmm... Yes and no...
From my point of view, I think it might that longer than we expect, maybe the entire month of May like you say.
BUT, they don't need to come to an "agreement", the HoC just needs to come to a decision about which amendments (if any at all) they accept and then the Senate will just almost automatically accept their decision. Like mentioned by many people (including Senators), it is extremely rare not to accept the HoC's final decision and send back a message to HoC "insisting" on amendments.