  The universal enveloping algebra of any semisimple Lie algebra $\mathfrak{g}$
contains a family of maximal commutative subalgebras, called shift of argument
subalgebras, parametrized by regular Cartan elements of $\mathfrak{g}$. For
$\mathfrak{g}=\mathfrak{gl}_n$ the Gelfand-Tsetlin commutative subalgebra in
$U(\mathfrak{g})$ arises as some limit of subalgebras from this family. In our
previous work (arXiv:1807.11126) we studied the analogous limit of shift of
argument subalgebras for the Lie algebras $\mathfrak{g}=\mathfrak{sp}_{2n}$ and
$\mathfrak{g}=\mathfrak{o}_{2n+1}$. We described the limit subalgebras in terms
of Bethe subalgebras of twisted Yangians $Y^-(2)$ and $Y^+(2)$, respectively,
and parametrized the eigenbases of these limit subalgebras in the finite
dimensional irreducible highest weight representations by Gelfand-Tsetlin
patterns of types C and B. In this note we state and prove similar results for
the last case of classical Lie algebras, $\mathfrak{g}=\mathfrak{o}_{2n}$. We
describe the limit shift of argument subalgebra in terms of the Bethe
subalgebra in the twisted Yangian $Y^+(2)$ and give a natural indexing of its
eigenbasis in any finite dimensional irreducible highest weight
$\mathfrak{g}$-module by type D Gelfand-Tsetlin patterns.
