Subregular coset ideal parametrization conjecture

For $n\geq4$, let $K_n$ be the ideal defining the subregular coset quotient and let $V(K_n)\subseteq\mathbb C^2$ be its affine variety in coordinates $(c,\lambda)$. Subregular coset parametrization conjecture. The variety $V(K_n)$ has the rational parametrization

$$c=-\frac{n(n^2-2k+nk-3n+1)(n^2-k+nk-2n-1)}{n+k},$$ $$\lambda=-\frac{(n+k-1)(n+k)}{(n^2-3k+nk-4n+2)(n^2-k+nk-2n-2)(n^2+k+nk)}.$$

The source states that this is equivalent to the subregular coset coincidence conjecture; no general proof is given.