Near-symmetry conjecture for generating functions of leafed-cycle cones of power-of-two length

Let $\mathcal{C}_{2^k}$ be the cone constrained by the Laplacian minor of a leafed $n$-cycle, where $n=2^k$ for some integer $k\geq 2$. Let $f(q)$ be the numerator in the proposed expression

$$\sigma_{\mathcal{C}_{2^k}}(q,1,1,\ldots,1)=\dfrac{f(q)}{\left(1-q^{2^k}\right)\displaystyle\prod_{i=0}^{k-1}\left(1-q^{2^{k-(i+1)}}\right)^{2^i}}.$$

Near-symmetry conjecture. The generating function has this form, and if $(a_0,\ldots,a_j)$ is the coefficient list of $f(q)$, then appending $0$ to the end of this list and subtracting its reverse produces the coefficient list of

$$\sum_{i=0}^{n-2}\binom{n-2}{i}(-1)^i q^{ni}.$$

This conjecture is based on experimental evidence for leafed cycles of length a power of two and describes the observed near-symmetry of the numerator despite the absence of a reflexive $n$-th slice in the non-prime case.