Symmetries of linear finite-type Frobenius systems

Let $\mu:\hat{Y}^{(n)} \subset \hat{X}^{(2n+2)}$ be a submanifold defined by an $n$-th level linear finite-type equation. Let $\hat{\rm I}^{(2n+2)}$ be the relevant Frobenius-system structure, and let $\textnormal{A}^k$ denote the Jacobi fields for $k\leq n$.

Symmetry conjecture. The pair $(\hat{Y}^{(n)},\mu^*\hat{\rm I}^{(2n+2)})$ is a Frobenius system. Moreover, each Jacobi field $\textnormal{A}^k$, $k\leq n$, has an associated locally defined symmetry vector field $V_{\textnormal{A}^k}$ on $\hat{Y}^{(n)}$, whose flow $\phi_k(t):\hat{Y}^{(n)}\to\hat{Y}^{(n)}$ is a symmetry of $\mu^*\hat{\rm I}^{(2n+2)}$.

The claim expresses the expected inheritance of the canonical Jacobi-field symmetries by the linear finite-type equation. The supplied text gives no resolution or further evidence, so its status remains open.