He's right. The flag has never been at half staff during a swearing in of president. . .except:

[onenote]

22. I'm not sure about all of those examples, but the Nixon example definitely is correct.

There have been numerous instances of presidents being sworn in within 30 days of the death of previous president, including all of the instances of a VP being sworn in upon the death of a sitting president. The latter group of swearing in ceremonies aren't really the kind of event most people think of as an "inauguration ceremony" and its not clear whether an order directing the flying of flags at half staff was made before or after the swearing in all of those cases.

But in the case of Nixon's 1973 inauguration, flags had been ordered flown at half staff following the death of Truman a few weeks earlier and were still at half staff on inauguration day:https://images.search.yahoo.com/search/images;_ylt=AwrFQJTeRXxnRKAJHWeJzbkF;_ylu=c2VjA3NlYXJjaARzbGsDYnV0dG9u;_ylc=X1MDOTYwNjI4NTcEX3IDMgRmcgN5ZnAtdARmcjIDcDpzLHY6aSxtOnNiLXRvcARncHJpZAMxRlVQUXZ6eFJ0bVZFODEzNC53NmhBBG5fcnNsdAMwBG5fc3VnZwMwBG9yaWdpbgNpbWFnZXMuc2VhcmNoLnlhaG9vLmNvbQRwb3MDMARwcXN0cgMEcHFzdHJsAzAEcXN0cmwDMzYEcXVlcnkDbml4b24lMjBpbmF1Z3VyYXRpb24lMjAxOTczJTIwcGhvdG9zJTIwZmxhZ3MEdF9zdG1wAzE3MzYxOTc2NDk-?p=nixon+inauguration+1973+photos+flags&fr=yfp-t&fr2=p%3As%2Cv%3Ai%2Cm%3Asb-top&ei=UTF-8&x=wrt#id=44&iurl=https%3A%2F%2Fmedia.gettyimages.com%2Fid%2F517285048%2Fphoto%2F1-20-1973-washington-dc-president-richard-nixon-inauguration-general-view-of-swearing-in.jpg%3Fs%3D612x612%26w%3Dgi%26k%3D20%26c%3DT872GPryj0aMa7f-NJXLeS75ld1vMvTK5M4jO13BBl8%3D&action=click