The only issue Thompson ever got excited by in the Senate was the McCain/Feingold bill. He was tepid about pro-life. How much confidence would you have that he would nominate strict constructionists if elected? Rudy, on the other hand, despite his rhetoric, has said over and over he would nominate strict constructionists in the mold of Roberts and Alito.
And in Guiliani's tortured logic, a "strict constructionist" could just as easily uphold Roe v. Wade as overturn it. That has to make you wonder what he thinks the term "strict constructionist" means.
I'm not seeing any reason to doubt him. In case you didn't notice, the President let McCain-Feingold go through his office with his signature on it...and then nominated two models of strict constructionism to the Court.