The bihamiltonian structures of the DR/DZ hierarchies at the approximation up to genus one

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.