  In a recent paper, giving an arbitrary homogeneous cohomological field theory
(CohFT), Rossi, Shadrin, and the first author proposed a simple formula for a
bracket on the space local functionals that conjecturally gives a second
Hamiltonian structure for the double ramification hierarchy associated to the
CohFT. In this paper we prove this conjecture at the approximation up to genus
1 and relate this bracket to the second Poisson bracket of the Dubrovin-Zhang
hierarchy by an explicit Miura transformation.
