My intuition would be that the evidence in favour of the 5, given by the fact you chose not to reveal them, doesn't come close to outweighing the 3-to-1 advantage granted by sticking.

Let me check with the actual calculations. If I stick I have a 3/12 chance of winning. If I switch and I originally did pick the winning cup then I always lose, and if I didn't then I win with probability 1/5. So my overall probability of winning if I switch is 3/12×0 + 9/12×1/5. So sticking gets me 1/4 and switching gets me 3/20. Sticking is better, but by a smaller margin than I expected.