Does Barker's conjectured order-10 recurrence for A321614 (maximum kings on a $4\times 2n$ board, free count) hold beyond the 22-term b-file?

OEIS A321614 counts placements of $2n$ nonattacking kings on a $4 \times 2n$ chessboard attaining the maximum $2n$, counted up to the rectangle's symmetry group (Burnside over identity, horizontal flip, vertical flip, $180°$ rotation). The published b-file stops at $n=21$ ($a(21)=665017743665$). From those 22 terms, Colin Barker (2018) conjectured an explicit order-10 linear recurrence $a(n) = 12a(n{-}1) - 54a(n{-}2) + 98a(n{-}3) + 17a(n{-}4) - 346a(n{-}5) + 505a(n{-}6) - 210a(n{-}7) - 120a(n{-}8) + 126a(n{-}9) - 27a(n{-}10)$ with matching rational generating function — an order-10 fit resting on only $\sim 11$ confirming terms, a genuinely thin margin. Kill or confirm: extend the sequence by orders of magnitude with certified, independently cross-checked code; report whether the minimal linear recurrence of the extended data is exactly Barker's (confirm, and establish minimality over the computed range) or the fit breaks at some $n$ (kill, with the first failing index).

Acceptance. FULLY RESOLVES: (1) reproduce ALL published b-file terms exactly (regression gate); (2) extend to at least $10\times$ the published index frontier with correctness certified by independent cross-checks (at minimum: a structurally different second implementation on a subrange, plus modular-arithmetic recomputation on spot indices over $\ge 2$ primes); (3) report the Berlekamp–Massey minimal recurrence of the extended sequence and whether it equals Barker's order-10 exactly; (4) all code and term data public and re-runnable. STRONGER: a proof of the recurrence for all $n$ from the transfer structure (e.g. the stabilized optimal-path subsystem of the max-plus DP). PARTIAL: a smaller extension ($\ge 2\times$ frontier) with the same certification, or a kill demonstrated with a verified first failing index.