Damped harmonic motion with one end without weight free

1. The problem statement, all variables and given/known data
A block on a horizontal surface is attached to two springs whose other ends are fixed to walls. A light string attached to one side of the block initially lies straight across the surface. The other end of the string is free to move. There is significant friction between the block and the surface but negligible friction between the surface and the string. The block is displaced in the direction of one of the walls and released from rest. What is the shape of the string a short time later?
That is: a top view is:
Wall
Spring
Weight_______________________ (string)
Spring
Weight

This is a conceptual, not a numerical, question, so that the choices would be the string’s amplitude constant, decreasing from left to right, increasing from left to right, or first increasing then decreasing.

2. Relevant equations

Since it is a conceptual problem, I don’t think that the equations are relevant.

3. The attempt at a solution
It is clear that the amplitude of the block’s motion decreases with time, and so as the wave travels from left to right, so should its amplitude. This therefore would, in my view, give the form of the string’s amplitude increasing from left to right. The fly in the ointment comes from the arrangement with an open-ended string. All treatments of damped harmonic motion that I find are either dealing with the amplitude of the block, or of a string with one end fixed (or at least not free: having a weight at the end). Therefore I am not sure whether this might cause the end configuration of the string to have an amplitude that first increases then decreases.

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