The reversibility conjecture for restricted integrability equations

Let ${\bf W}$ be a component of the integrability variety ${\bf V}(B)$ of the system under consideration. Let $\mathcal A|_{\bf W}$ denote the restriction to ${\bf W}$ of the operator in equation (e:AonW), and let $\widetilde V(t_1,\dots,t_m)=1+\operatorname{h.o.t.}$ be a formal power series. Reversibility conjecture. If equation $\mathcal A|_{\bf W}(\widetilde V)=0$ has a formal power series solution $\widetilde V(t_1,\dots,t_m)=1+\operatorname{h.o.t.}$, then the original equation $\mathcal A(V)=0$ has a solution $V$ such that $V|_{\bf W}=1+\operatorname{h.o.t.}$. The claim is presented as the converse of the preceding restriction theorem; the supplied text gives no resolution status.