Initially objected then WITHDREW their objection and refused to join HOUSE members in writing. Not ONE US Senator SIGNED an objection with the HOUSE and therefore the HOUSE objections were NOT entertained and the vote was counted.
In four of the original six states mentioned for objections then no, senators did not sign on. But for Arizona and Pennsylvania there were Senators who joined House members in objecting, and there were debates and votes on the objections as a result. In fact they were in the process of adjourning to separate chambers for the Arizona debate when the Capitol was breached.