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Liberals will always have a public relations advantage over conservatives, because Liberals can accuse Conservatives of immorality and that becomes a problem for the poor accused person. Conservatives generally believe, and generally state, that morality is a good thing.
On the other hand, Conservatives can't accuse Liberals of immorality because for Liberals, immorality is normal, no problem. Heck, to them it's admirable.
Put another way, both Conservatives and Liberals have character flaws. The only difference is that conservatives are ashamed of them and liberals are proud of them.