Combinatorial-interpretation conjecture for the Gessel sequence

Define the Gessel sequence by

$$b_n=2\cdot 5^n-(3+4i)^n-(3-4i)^n,$$

where $i=\sqrt{-1}$. A combinatorial interpretation of type (C1) is the type specified by the paper for a combinatorial explanation of a sequence.

Gessel-sequence interpretation conjecture. The sequence $b_n$ has a combinatorial interpretation of type (C1).

The sequence is C-recursive and nonnegative, but the source describes it as lacking an easy combinatorial interpretation. The claim is an open problem in the supplied text.