I have noticed experimentally that the following question has a positive answer.

Is it true that for all $f, g$ even and convex functions :

$$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x) dx \times \int_0^1 g(\cos(1/x))dx? $$