A conjectural formula for $D{R}_g(a,-a){\lambda }_g$

We propose a conjectural formula for $D{R}_g(a,-a){\lambda }_g$ and check all its expected properties. Our formula refines the one point case of a similar conjecture made by the first named author in collaboration with Gu'er'e and Rossi, and we prove that the two conjectures are in fact equivalent, though in a quite non-trivial way.