Unbounded pole order

Prove that the pole orders of F_{ω^(k)}(t)/(1−t^2), where ω^(k) consists of k parts equal to 3 separated consecutively by two parts equal to 1, are unbounded as k varies, equivalently proving that the eventual quasi-polynomial degrees of |χ_d(ω^(k))| are unbounded.

Background

Repeated copies of the pattern formed by a part 3 followed by two parts 1 produce increasingly large Euler characteristics. The computed eventual quasi-polynomial degrees are 1, 3, and 6 for k=2, 3, and 4, respectively.

These calculations suggest that the complexity of the Euler-characteristic growth increases without bound as the repeated pattern grows, but no general proof is given.