The truncation-curve conjecture for the universal even spin b[W]_-infinity algebra

Let $n\geq 2$ and let $\mathcal F^k(n)$ be the universal vertex algebra considered in the paper. A truncation curve in the $(c,\lambda)$-plane is parametrized by

$$c_n(k)=-\frac{(1+2k)(2+2k+n)(2+3k+2n)}{(1+k+n)(1+2k+2n)},$$ $$\lambda_n(k)=-\frac{(1+k+n)(1+2k+2n)p_n(k)}{7(1+k)(1+2k+n)(5+6k+4n)q_n(k)r_n(k)},$$

where

$$p_n(k)=-60-306k-408k^2+198k^3+720k^4+360k^5-294n-812kn+177k^2n+1916k^3n+1236k^4n-360n^2-153kn^2+1606k^2n^2+1504k^3n^2-102n^3+464kn^3+776k^2n^3+24n^4+144kn^4,$$ $$q_n(k)=6+9k+6k^2+15n+20kn+12n^2,$$ $$r_n(k)=-2+24k+86k^2+60k^3-36n+7kn+70k^2n-34n^2+20kn^2.$$

Let $K_n\subseteq\mathbb C[c,\lambda]$ be the ideal corresponding to this curve. The truncation-curve conjecture. The quotient

$$\mathcal W^{\mathrm{ev},K_n}(c,\lambda)=\mathcal W^{\mathrm{ev}}(c,\lambda)/(K_n\cdot\mathcal W^{\mathrm{ev}}(c,\lambda))$$

has a singular vector of weight $2n^2+2n$, and, after a suitable localization, its simple quotient is isomorphic to $\mathcal F^k(n)$. This gives an explicit rational parametrization of the truncation curve realizing $\mathcal F^k(n)$ as a quotient of $\mathcal W^{\mathrm{ev}}(c,\lambda)$.