The open KdV conjecture for open gravitational descendents

Let $F^o(s,t_0,t_1,\ldots;u)$ be the open free energy, with double brackets denoting its derivatives with respect to the variables $t_i$ and $s$, and let $F^c$ denote the closed free energy. Open KdV conjecture. For every $n\mathbf{e} 1$, the following equations are satisfied:

$$(2n+1)u^{-1}{\big\langle}{\big\langle} \tau_n {\big\rangle}{\big\rangle}^o=u{\big\langle}{\big\langle} \tau_{n-1} \tau_0{\big\rangle}{\big\rangle}^c {\big\langle}{\big\langle} \tau_0{\big\rangle}{\big\rangle}^o -\frac{u}{2} {\big\langle}{\big\langle} \tau_{n-1}\tau_0^2 {\big\rangle}{\big\rangle}^c+2 {\big\langle}{\big\langle} \tau_{n-1}{\big\rangle}{\big\rangle}^o {\big\langle}{\big\langle}\sigma{\big\rangle}{\big\rangle}^o + 2 {\big\langle}{\big\langle} \tau_{n-1} \sigma {\big\rangle}{\big\rangle}^o.$$

The open string and open dilaton equations are known in genus zero and, by the cited context, in all genera; the main open KdV system itself was proved in the referenced later work.