They have to be signing in corporate character as they are not the de jure Senate since 1914. The corporate takeover by impersonation was complete by then with the adoption of the Federal Reserve, 16th Amendment taxes, and the 17th Amendment changes to the Senate.
"8th: Accordingly, in 1914, the Freshman class and all Senators that successfully ran for reelection in 1913 by popular vote were seated in Corp. U.S. Senate capacity only; respectively, the original jurisdiction Senate seats from their respective States remained vacant; because, neither the State Senates nor the State Governors appointed new original jurisdiction Senators to replace them; which is still required by the national Constitution for placement of a national government Senator."
https://teamlaw.net/HistoryOutline.htm