I found out that my claim works only when p!=n (Actually the original paragraph also turns out to have this assumption, but it is not explicitly written), so I propose changing the last paragraph wholly.

Now let us assume that  $n$  is not divisible by  $k$ . We show that this implies that the length of the answer is  $n$ . We prove it by contradiction. Assuming there exists an answer, and the compression has length  $p$  ( $p$  divides  $n$, $p \neq n$ ). Then the last value of the prefix function has to be greater than  $n - p$ , i.e. the suffix will partially cover the first block. Now consider the second block of the string. Since the prefix is equal with the suffix, and both the prefix and the suffix cover this block and their displacement relative to each other  $k$  does not divide the block length  $p$  (otherwise  $k$  divides  $n$ ). But then the string consists of $gcd(p, k)$ characters repeated over and over, hence we can compress it to a string of size $gcd(p, k)$, and since this value divides $p$ it also divides $n$. Contradiction.

I can't see a contradiction to the initial assumption that k doesn't divide n. I might be missing something.

Ah, now I see that my argument was a little bit wrong. I intended to show a contradiction in another way but I got mixed up. Now I change the comment a little.

Now let us assume that  $n$  is not divisible by  $k$ . We show that this implies that the length of the answer is  $n$ . We prove it by contradiction. Assuming there exists an answer, and the compression has length  $p$  ( $p$  divides  $n$, $p \neq n$ ). Then the last value of the prefix function has to be greater than  $n - p$ , i.e. the suffix will partially cover the first block. Now consider the second block of the string. Since the prefix is equal with the suffix, and both the prefix and the suffix cover this block and their displacement relative to each other  $k$  does not divide the block length  $p$  (otherwise  $k$  divides  $n$ ). But then the string consists of $gcd(p, k)$ characters repeated over and over, hence we can compress it to a string of size $gcd(p, k) < p$. But this contradicts the definition of $p$.

Ah I see, would you need to add the definition/condition that p is selected to be the smallest possible concatenation unit length?

Yes, you are right in pointing that out.
BTW, another way of leading a contradiction is stating that gcd(p,k)<k and the block length gets smaller than n - (last prefix function value), which leads to a contradiction.
If you are going to accept my fix suggestion, I think either way is fine and you may choose.(stating that p is minimum, or using the prefix function's definition to show contradiction.)