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Determine maximum frequency of input signal to make system LTI

This is question 4.26 from the third edition of Alan Oppenheim's textbook "Discrete-time Signal Processing". It is stated as follows: enter image description here

Firstly, the answer is the input signal should be bandlimited to 12 kHz. I'm having trouble seeing why it is the case.

My reasoning:

The cutoff frequency of the low pass filter is obviously 8 kHz, so any frequency component of the discretized input signal beyond 8 kHz will be removed. But since we do not want aliasing to occur, we want the input signal to be bandlimited such that after passing the sampled signal through the low pass filter we don't get aliasing below 8 kHz. But in the case of input signal bandlimited to 12 kHz, we get the following (a quick drawing): enter image description here

which causes aliasing, making the overall system no longer LTI.

So where did I go wrong? Any help is greatly appreciated!