  We present a family of conjectural relations in the tautological cohomology
of the moduli spaces of stable algebraic curves of genus $g$ with $n$ marked
points. A large part of these relations has a surprisingly simple form: the
tautological classes involved in the relations are given by stable graphs that
are trees and that are decorated only by powers of the psi-classes at
half-edges. We show that the proposed conjectural relations imply certain
fundamental properties of the Dubrovin-Zhang (DZ) and the double ramification
(DR) hierarchies associated to F-cohomological field theories. Our relations
naturally extend a similar system of conjectural relations, which were proposed
in an earlier work of the first author together with Gu\'er\'e and Rossi and
which are responsible for the normal Miura equivalence of the DZ and the DR
hierarchy associated to an arbitrary cohomological field theory. Finally, we
prove all the above mentioned relations in the case $n=1$ and arbitrary $g$
using a variation of the method from a paper by Liu and Pandharipande, this can
be of independent interest. In particular, this proves the main conjecture from
our previous joined work together with Hern\'andez Iglesias. We also prove all
the above mentioned relations in the case $g=0$ and arbitrary $n$.
