Dušek's $k=0$ conjecture for null homogeneous geodesics on g.o. spaces

Let $G/H$ be a pseudo-Riemannian homogeneous geodesic orbit space. A homogeneous geodesic is an orbit satisfying the geodesic lemma, whose parameter is governed by a scalar $k$; for a null homogeneous geodesic, consider this scalar in the geodesic lemma.

Dušek's $k=0$ conjecture. For all null homogeneous geodesics in $G/H$, one has $k=0$ in the geodesic lemma.

The paper's example is almost g.o. but not g.o., and therefore does not settle this restricted geodesic-orbit formulation. The authors state that the conjecture remains open for homogeneous g.o. spaces.