Exact degree of regularity of the nonlinear equation L_n

Determine the exact degree of regularity of the nonlinear equation \(L_n=\sum_{i=1}^{n-1}2^{2i}X_i^{2i}-X_n^{2n}-2^{2n}X_{n+1}X_{n+2}=0\) for even positive integers \(n\).

Background

Corollary 4.1 establishes bounds on the degree of regularity of the displayed nonlinear equation when $n$ is even: it is at least $n-1$ and at most $n$. The authors explicitly note that the exact degree is unknown, so the remaining problem is to decide which of these two values is attained.