Variational characterization conjecture for additive difference equations

Let $k$ be a positive integer, and consider an additive $2k$th-order difference equation

$$x_{n+k}=f\bigl(\mathbf{x}_{n}^{(-k+1,k-1)}\bigr)x_{n-k}+h\bigl(\mathbf{x}_{n}^{(-k+1,k-1)}\bigr).$$

Here $\mathbf{x}_{n}^{(m,l)}=(x_{n+m},\dots,x_{n+k})$ as defined in the source, and $f$, $h$, $V$, and $\lambda$ are the functions and constant occurring in the equation and Lagrangian. Variational characterization conjecture. The additive $2k$th-order difference equation is variational if and only if it can be derived from the Lagrangian

$$L_{n}^{(k)}=\lambda^{-n}\left[f\bigl(\mathbf{x}_{n}^{(1,k-1)}\bigr)x_nx_{n+k}+V\bigl(\mathbf{x}_{n}^{(0,k-1)}\bigr)\right].$$

The conjecture seeks a complete characterization of variational additive difference equations in every even order. The paper proves the corresponding fourth-order result, but gives no resolution of the proposed higher-dimensional statement.