Two-line-family conjecture for the standard monomials of the Fibonacci Lie algebra

Let $W_N$ be the set of standard monomials of length $N$ in the Fibonacci Lie algebra, and regard their multidegrees as points in the plane. Two-line-family conjecture. For every fixed $N\ge 1$, the monomials in $W_N$ belong to two families of parallel lines. This is a geometric conjecture suggested by the computed Hilbert-series data and figures; a general proof is not supplied.