quantum mechanics - Why doesn't $σ_xσ_p$ change with the width of the well in the infinite square well problem (intuition)?

I calculated that the product of the uncertainty in position $\sigma_x$ for the ground state of an infinite square well of width $L$ with the uncertainty in the momentum $\sigma_p$ for the same state, and it does not change with respect to the width of the well $L$. Intuitively, why does this happen?

(I would also like to know what the physical reasons are for the decrease in the momentum uncertainty, but I have asked that separately.)