In the photocurrent experiment, if the intensity of light is kept constant, but the frequency is slightly increased, what's the effect on the saturation current?

I think saturation current should decrease because the number of photons should decrease.

But, all my reference books say that the saturation current depends only on intensity, and since it is constant, hence there will be no change.

Can you please tell me which reasoning is correct? And why?

In the photocurrent experiment, if the intensity of light is kept constant, but the frequency is slightly increased, what's the effect on the saturation current?

I think saturation current should decrease because the number of photons should decrease.

But, all my reference books say that the saturation current depends only on intensity, and since it is constant, hence there will be no change.

Can you please tell me which reasoning is correct? And why?

EDIT: Another confusing factor is the energy of electrons. If the electrons have more energy due to the increased frequency then I would take less time to get to the anode and that should increase the current. Now, which factor is more dominant? And how do i know that?