Kawasetsu's explicit ideal conjecture for minimal W-algebra cosets

For $n\geq4$, let $K_n$ be the ideal in $\mathbb C[c,\lambda]$ defining the two-parameter quotient associated with the minimal $\mathcal W$-algebra coset. Kawasetsu's explicit ideal conjecture. The ideal $K_n$ is generated by the polynomial

$$\begin{aligned} p_n(c,\lambda)={}&3+108\lambda-12c\lambda+1152\lambda^2-384c\lambda^2+3072\lambda^3-3072c\lambda^3-4n-144\lambda n+20c\lambda n\\ &-1536\lambda^2n+592c\lambda^2n-16c^2\lambda^2n-4096\lambda^3n+4352c\lambda^3n-256c^2\lambda^3n-2n^2-28\lambda n^2\\ &-2c\lambda n^2+100\lambda^2n^2-152c\lambda^2n^2+4c^2\lambda^2n^2+576\lambda^3n^2-1920c\lambda^3n^2+192c^2\lambda^3n^2\\ &+4n^3+94\lambda n^3-8c\lambda n^3+548\lambda^2n^3-144c\lambda^2n^3+4c^2\lambda^2n^3+1088\lambda^3n^3+32c\lambda^3n^3\\ &+32c^2\lambda^3n^3-n^4-32\lambda n^4+2c\lambda n^4-199\lambda^2n^4+80c\lambda^2n^4-c^2\lambda^2n^4-384\lambda^3n^4\\ &+144c\lambda^3n^4-48c^2\lambda^3n^4+2\lambda n^5+4\lambda^2n^5-10c\lambda^2n^5-16\lambda^3n^5+8c\lambda^3n^5+8c^2\lambda^3n^5\\ &+3\lambda^2n^6+12\lambda^3n^6-12c\lambda^3n^6. \end{aligned}$$

The corresponding variety is parametrized by $c=-\frac{(1+k)(2k+n-1)(3k+2n)}{(k+n-1)(k+n)}$ and $\lambda=\frac{(k+n-1)(k+n)}{(n-2)(2k+n-2)(4k+3n)}$. The conjecture is verified by computer when $n=4$ and is equivalent to Kawasetsu's coset coincidence conjecture.