  The Kundt conjecture states that a Lorentzian manifold of arbitrary dimension
which is not characterized by its scalar polynomial curvature invariants (SPIs)
allows for a non-twisting, non-shearing and non-expanding (in short, Kundt)
null congruence of geodesics. The conjecture has been proven for dimensions 3
and 4. A necessary condition for a spacetime not to be characterized by SPIs is
that all covariant derivatives of the Riemann tensor are of aligned type II or
more special in the null alignment classification. In arbitrary dimensions, we
prove that this property indeed requires the presence of a Kundt null
congruence when a certain genericity condition holds, or when the trac-free
Ricci orWeyl tensor is of genuine type III or N, thus confirming the validity
of the Kundt conjecture in these cases. We also strenghten the results for
dimensions 3 and 4 by removing regularity assumptions and showing that only the
third covariant derivative is needed to obtain the results. A key tool in our
proofs is a new bilinear map acting on tensors of related boost orders relative
to a null direction.
