  We study the Witt classes of the modular categories $SO(2r)_{2r}$ associated
with quantum groups of type $D_r$ at $4r-2$th roots of unity. From these
classes we derive infinitely many Witt classes of order 2 that are linearly
independent modulo the subgroup generated by the pointed modular categories. In
particular we produce an example of a simple, completely anisotropic modular
category that is not pointed whose Witt class has order 2, answering a question
of Davydov, M\"uger, Nikshych and Ostrik. Our results show that the trivial
Witt class $[Vec]$ has infinitely many square roots modulo the pointed classes,
in analogy with the recent construction of infinitely many square roots of the
Ising Witt classes modulo the pointed classes constructed in a similar way from
certain type $B_r$ modular categories. We compare the subgroups generated by
the Ising square roots and $[Vec]$ square roots and provide evidence that they
also generate linearly independent subgroups.
