The Smith normal form of the walk matrix of the Dynkin graph $D_n$ for $n\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$

Let $W(D_n)$ denote the walk matrix of the Dynkin graph $D_n$. We prove that the Smith normal form of $W(D_n)$ is $$\text{diag}[\underset{\frac{n}{2}-1}{\underbrace{1,1,\dots ,1}},\underset{\frac{n}{2}-1}{\underbrace{2,2,\dots ,2}},0,0]$$ when $n\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$. This gives an affirmative answer to a question in [W. Wang, C. Wang, S. Guo, On the walk matrix of the Dynkin graph $D_n$, Linear Algebra Appl. 653 (2022) 193–206].