Kernel non-triviality conjecture for tautological classes of strata of differentials

Let $g\geq 2$, let $\ell$ be a positive integer, and let $\mu=(m_1,\ldots,m_k)$ be a partition of $\ell(2g-2)$ with $m_i\neq-\ell$ for every $i$. Let $\mathcal{P}^\ell(\mu)$ be the associated stratum and let the map in equation (t1) be the tautological-class map whose kernel is under consideration. Kernel non-triviality conjecture. The kernel of this map is non-trivial in degree

$$* = \lfloor g/3\rfloor+1.$$

This is the precise general form of the conjectured non-triviality of relations obtained by pulling back known relations from $R^*(\mathcal{M}_{g,n})$. The restriction $m_i\neq-\ell$ is essential in the supplied text, since the pullback is trivial when some $m_i=-\ell$; the conjecture is otherwise unresolved, apart from the cases established in the paper.