On the walk matrix of the Dynkin graph $D_n$

Let $W(D_n)$ denote the walk matrix of the Dynkin graph $D_n$, a tree obtained from the path of order $n-1$ by adding a pendant edge at the second vertex. We prove that $\text{rank}\,W(D_n)=n-2$ if $4\mid n$ and $\text{rank}\,W(D_n)=n-1$ otherwise. Furthermore, we prove that the Smith normal form of $W(D_n)$ is $$\text{diag}[\underbrace{1,1,\ldots,1}_{\lceil\frac{n}{2}\rceil},\underbrace{2,2,\ldots,2}_{\lfloor\frac{n}{2}\rfloor-1},0]$$ when $4\nmid n$. This confirms a recent conjecture in [W.Wang, F.Liu, W.Wang, Generalized spectral characterizations of almost controllable graphs, European J. Combin., 96(2021):103348].