Measuring momentum in a 2 height well

In a 2 height finite square well we know that qualitatively the position wavefunction will have a lower amplitude in the side of the well with a smaller potential energy (since it will be travelling faster and spend less time on average there), but we also know that qualitatively it will have a higher average wavelength in this side of the well (since it will have a higher average momentum here).

Does that mean that if we measure (not exactly but with some uncertainty) the position and momentum of our particle, there will be a correlation of higher momentum for the side of the well with lower potential energy?

I’m not certain since if we decompose our wavefunction into its respective momentum wavefunctions they should be equally valid ‘everywhere’.

Although maybe in order to calculate what we would measure in our small measurement region we would have to reformulate the problem anyway?

Which thinking is more correct?