Griffiths transversality for variational conservation laws of constant mean curvature surfaces

For a first-order deformation determined by the inhomogeneous Jacobi field $u=\mathrm{i}v_{\rho}+\mathrm{i}\delta v_0$, let $\psi_j$, $j=0,1,\ldots$, be the sequence of $1$-forms obtained by varying the higher-order conservation law $\varphi_j$. Griffiths transversality. As a characteristic cohomology class on $\hat{Z}$, one has

$$[\psi_j]\in\langle[\varphi_0],[\varphi_1],\ldots,[\varphi_{j-2}]\rangle\subset H^1(\Omega^*(\hat{Z})/\hat{\mathcal J},\underline{\mathrm{d}}).$$

This conjecture predicts that the variational conservation laws introduce no new characteristic cohomology classes on $\hat{Z}$: each $[\psi_j]$ is a linear combination of higher-order conservation-law classes of lower index. The preceding computations establish the claim for the first few terms, but the general statement remains open.