The structure conjecture for the algebras $\mathcal V^{\mathbb{C},\lambda,n}$

Let $\mathcal V^{\mathbb{C},\lambda,n}$ be the algebra under consideration, and let $\bar s,t,\lambda$ be variables of formal degrees $2,1,-2$, respectively. Define the formal series $\bar f_k(\bar s,t,\lambda)$ by

$$\sum \bar f_k(\bar s,t,\lambda)x^k=\log\left(1+\bar s x^2+t x+\lambda x^{-2}+3\lambda^2x^{-4}+13\lambda^3x^{-6}+\cdots\right)$$

and equivalently by

$$\log\left(1+\bar s x^2+t x+\sum_{n\geq 1}\left[\binom{4n+1}{n+1}-9\binom{4n+1}{n-1}\right]\lambda^n x^{-2n}\right).$$

The structure conjecture. For every $i>n$, one has

$$\bar f_i(\bar s,t,\lambda)=0$$

in $\mathcal V^{\mathbb{C},\lambda,n}$. This is presented as the first conjecture concerning the structure of these algebras and is motivated by computations and the Online Encyclopedia of Integer Sequences. The supplied text does not establish the claim or provide evidence of its resolution.