Question

Rabbi Joseph Dov Soloveitchik was reported near the end of his life as saying, “I never signed a permit of 100 rabbis.”
However, he did not instruct others to refrain from doing so, and his students did sign and still do sign, and he himself said that he refrained only because of the desecration of God’s name involved in it—because people would say…
But not because he saw any flaw in the permit of 100 rabbis itself.
 
Is it proper for a God-fearing rabbi not to sign because of that same concern?
Or is this unique to someone who was a great man, or one of the greatest Jewish philosophers, such that if he were to sign it there would be a greater desecration of God’s name, since such a distinguished and cultured person would be participating in such an injustice? But for a town rabbi or neighborhood rabbi who is not well known or especially esteemed, there is no concern of desecration of God’s name?