The first 26-tuple was found in the project

5179852391836338871: 0 12 18 28 46 76 78 120 186 210 226 232 238 300 306 312 328 352 418 460 462 492 510 520 526 538

https://boinc.termit.me/adsl/spt_list.php?k=26

See https://oeis.org/A055382

Demis reported: "the author of find k=26 is [AF] Kalianthys".

Read more in Russian
https://boinc.progger.info/odlk/forum_thread.php?id=236&postid=12248

We are waiting for confirmation of the minimality of the found tuple in order to write it to OEIS.