Delayed-Fibonacci torsion conjecture for the four-element Alexander quandle

Let $X=\mathbb{Z}_2[t]/(1+t+t^2)$ be the four-element Alexander quandle, and let $\operatorname{H}^{\rm Q}_n(X)$ denote its quandle homology. Let ${f_n}_{n\ge1}$ be the sequence of delayed Fibonacci numbers, and define

$$f'_n=\log_2\left(\left|\operatorname{Tor}\operatorname{H}^{\rm Q}_n(X)\right|\right)-2f_n.$$

Delayed-Fibonacci torsion conjecture. The torsion subgroup of $\operatorname{H}^{\rm Q}_n(X)$ is isomorphic to $\mathbb{Z}_4^{f_n}\oplus\mathbb{Z}_2^{f'_n}$. The displayed low-degree computations motivate this pattern, but no proof of the general formula is given in the supplied text.