Here's an example which doesn't bear on Conservation of Expected Evidence as math, but does bear on the statement,

"There is no possible plan you can devise, no clever strategy, no cunning device, by which you can legitimately expect your confidence in a fixed proposition to be higher (on average) than before."

taken at face value.

It's called the Cable Guy Paradox; it was created by Alan Hรกjek, a philosopher the Australian National University. (I personally think the term Paradox is a little strong for this scenario.)

Here it is: the cable guy is coming tomorrow, but cannot say exactly when. He may arrive any time between 8 am and 4 pm. You and a friend agree that the probability density for his arrival should be uniform over that interval. Your friend challenges you to a bet: even money for the event that the cable guy arrives before noon. You get to pick which side of the bet you want to take -- by expected utility, you should be indifferent. Here's the curious thing: if you pick the morning bet, then almost surely there will be times in the morning when you would prefer to switch to the afternoon bet.

This would seem to be a situation in which "you can legitimately expect your confidence in a fixed proposition to be higher (on average) than before," even though the equation P(H) = P(H|E)*P(E) + P(H|~E)*P(~E) is not violated. I'm not sure, but I think it's due to multiple possible interpretations of the word "before".