Suppose the object just touches two spring(not attached) and then is displaced towards left by some distance and released then I have seen that this mass will undergo simple harmonic motion on both sides with spring constant k1 and k2 on right on left and right side respectively
Now consider the case when the springs are attached to the block and then it is displaced in the same manner.
Consider the first case where it is displaced towards left by some distance and then released.Then it will move towards right due to spring force and then it will reach the equilibrium point .At this point ,it will be touching the springs so no spring force is applied and then it will feel spring force from the right wall as it moves and then it again comes back.
During all those motion ,How is amplitude in both directions conserved in this case. when they are subjected to different forces on either side on this case and even on the attached case