Buryak–Rossi–Shadrin bihamiltonian conjecture for the DR hierarchy

Let $V$ be the state space of a homogeneous cohomological field theory with homogeneous basis $e_1,\ldots,e_N$, metric $(\eta^{\alpha\beta})$, degrees $\deg e_\alpha=q_\alpha$, and parameters $r^\gamma$ defined by $\overline r=r^\gamma e_\gamma$. Set

$$\mu=\operatorname{diag}(\mu_1,\ldots,\mu_N),\qquad \mu_\alpha=q_\alpha-\frac{\delta}{2},$$

and let $K^\mathrm{DR}$ be the matrix differential operator defined from the DR hierarchy. Define

$$A^\beta_\alpha:=\eta^{\beta\nu}c_{0,3}(e_\nu\otimes e_\alpha\otimes\overline r).$$

For the DR Hamiltonians $\overline g_{\alpha,d}$, let $\{\cdot,\cdot\}_{K^\mathrm{DR}}$ and $\{\cdot,\cdot\}_{\eta^{-1}\partial_x}$ denote the brackets induced by $K^\mathrm{DR}$ and the first Poisson operator $\eta^{-1}\partial_x$, respectively.

Buryak–Rossi–Shadrin's conjecture. The operator $K^\mathrm{DR}$ is Poisson and compatible with $\eta^{-1}\partial_x$. It endows the DR hierarchy with a bihamiltonian structure satisfying

$$\left\{\cdot,\overline g_{\alpha,d}\right\}_{K^\mathrm{DR}}=\left(d+\frac{3}{2}+\mu_\alpha\right)\left\{\cdot,\overline g_{\alpha,d+1}\right\}_{\eta^{-1}\partial_x}+A^\beta_\alpha\left\{\cdot,\overline g_{\beta,d}\right\}_{\eta^{-1}\partial_x},\qquad d\geq -1.$$

This conjecture proposes a second compatible Poisson structure for the DR hierarchy, making it bihamiltonian and providing a recursion among its Hamiltonian flows. The supplied text does not state whether it has been proved or disproved.