I am trying to better understand this nice answer to a question of mine, which states

Spin structures on a compact complex manifold $(M^{2n},J)$ are in bijective correspondence with isomorphism classes of holomorphic line bundles ${\cal L}$ such that ${\cal L}\otimes {\cal L} = {\cal K}$ where ${\cal K}$ is the canonical line bundle of $(M,2n)$.

Now in an answer to another question on the canonical bundle of the Lagrangian Grassmannians, it is stated that

(paraphrased) The Picard group of each Lagrangian Grassmannian $Sp(2n)/P$ (where $P$ is the maximal parabolic subgroup) is cyclic with generator ${\cal O}(-1)$. Moreover, its canonical bundle is $\cal{O}(-n-1)$.

Now since the Lagrangian Grassmannian is a flag manifold, and hence a complex manifold, these two facts seem to me to imply that

The Lagrangian Grassmannian is a spin manifold if and only if $n$ is odd.

Is my reasoning correct?