An improvement in what? Money/fame yeah, but not playing levels. Lazio's keeper coaches are some of the best in the world, and he's benefited plenty from them. Moving from a team that's only marginally better, with shittier coaches, is actually more detrimental than anything. I don't know anything about Liverpool's keeper coaches but their keeper situation has been abysmal.
If he was an outfield player, then playing under Klopp would be beneficial, but Klopp will not help him in any way, shape or form.