Dual dilaton with $\mathcal{R}$ and $\mathcal{Q}$ fluxes

In previous works we showed that a Courant algebroid in a particular frame and the differential geometry of the sum bundle $TM \oplus T^*M$ provide a very natural geometric setting for a sector of the low energy effective limit of type II superstring theories (Supergravity theory). Given our geometric and algebraic considerations, we reproduced the NS-NS sector of the closed bosonic effective type II sting action, and an action for the inverse metric $G^{-1}$ and the bivector $\Pi$, related to the tensors for closed strings as $(g+B)^{-1} = (G^{-1} +\Pi)$. The action depended on the stringy T-dual fluxes $\mathcal{R}$ and $\mathcal{Q}$, but the dual dilaton was missing. This short paper fills the gap.