A recurrent relation for the oscillator specialization of the c(2)-weight system

Suppose that $\xi(G)$ is a 4-invariant of graphs extending the specialization of the $\mathfrak{sl}(2)$-weight system at $c=-\frac{3}{32}$. Let $G$ be a graph, let $x$ be a vertex of degree $2$, and let $A=\{a_1,a_2\}$ be its neighbors. Write $G_x$ for the graph obtained by deleting $x$, $\overline{G}_x$ for the graph obtained by deleting $x$ and inverting the edge $(a_1,a_2)$, and $G_x^{a_1}$ for the graph obtained from $G_x$ by applying the 2T transform to $(a_1,a_2)$ and deleting $a_1$. The recurrent relation conjecture. The following relation holds:

$$\xi(G)=\left(-\frac{3}{32}-1\right)\xi(G_x)-\frac{1}{2}\xi(\overline{G}_x)+\frac{1}{2}\cdot\left(-\frac{3}{32}\right)\xi(G_x^{a_1}).$$

This would provide a recurrence for the conjectural 4-invariant extending the $c=-\frac{3}{32}$ specialization, whose existence is motivated by the oscillator representation of $\mathfrak{sl}(2)$.