The invariance criterion for tautological equations

Let $R^k(\overline{\mathcal{M}}_{g,n})$ be the tautological classes of codimension $k$ on $\overline{\mathcal{M}}_{g,n}$. Let $E=\sum_i c_i\Gamma_i$ be a linear combination of codimension $k$ tautological strata in $\overline{\mathcal{M}}_{g,n}$, where $k<3g-3+n$ and the $c_i$ are unknown variables. For $l=1,2,\ldots$, let

$$\mathfrak{r}_l:R^k(\overline{\mathcal{M}}_{g,n})\to R^{k+l-1}(\overline{\mathcal{M}}^{\bullet}_{g-1,n+2})$$

be the operators on decorated graphs introduced in the source.

Invariance criterion. If

$$\mathfrak{r}_l(E)=0$$

for all $l$, then $E=0$ is a tautological equation.

This is the second invariance conjecture for tautological equations. The paper states that it is intended to produce all tautological equations inductively; the supplied text does not establish its general resolution.