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

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

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