Empirical recurrence conjecture for balanced 8 by 2n matrices

Let $b_4(n)$ denote the number of balanced $8\times 2n$ matrices. A linear recurrence of order $9$ with polynomial coefficients of degree $36$ should be satisfied by the sequence $b_4(n)$; the recurrence is too large to print and is available at the cited website. This recurrence was obtained empirically by fitting sequence terms and is presented as conjectured, though the authors describe it as absolutely certain.