Inductive generation of tautological equations

Let $E=\sum_i c_i\Gamma_i$ be a linear combination of codimension $k$ tautological strata in $\overline{\mathcal{M}}_{g,n}$, with $k<3g-3+n$ and unknown coefficients $c_i$. Let $\mathfrak{r}_l$ be the graph operations that lower the genus by one and add two marked points.

Inductive generation conjecture. Conjecture 2 will produce all tautological equations inductively.

The claim describes the intended consequence of the invariance criterion: applying the operators $\mathfrak{r}_l$ should generate every tautological equation through an inductive procedure. The supplied text gives no resolution of this assertion.