  In this paper we present a family of conjectural relations in the
tautological ring of the moduli spaces of stable curves which implies the
strong double ramification/Dubrovin-Zhang equivalence conjecture. Our
tautological relations have the form of an equality between two different
families of tautological classes, only one of which involves the double
ramification cycle. We prove that both families behave the same way upon
pullback and pushforward with respect to forgetting a marked point. We also
prove that our conjectural relations are true in genus $0$ and $1$ and also
when first pushed forward from $\overline{\mathcal{M}}_{g,n+m}$ to
$\overline{\mathcal{M}}_{g,n}$ and then restricted to $\mathcal{M}_{g,n}$, for
any $g,n,m\geq 0$. Finally we show that, for semisimple CohFTs, the DR/DZ
equivalence only depends on a subset of our relations, finite in each genus,
which we prove for $g\leq 2$. As an application we find a new formula for the
class $\lambda_g$ as a linear combination of dual trees intersected with kappa
and psi classes, and we check it for $g \leq 3$.
