special relativity - Why is it necessary that different observers agree on the value of the spacetime interval $ds^2$?

What's the physical reason that all (inertial) observers agree on the value of the spacetime interval

$$ds^2 = (c dt)^2 - dx^2 - dy^2 -dz^2 \, ?$$

What would be the physical implications if different (inertial) observers wouldn't find different values of $ds^2$, analogous to how they find different time intervals and different distances?

EDIT: none of the linked questions (to which this question supposedly is a duplicate) explain why a constant speed of light implies that all observers agree on the value of the spacetime interval.