Smith normal form conjecture for the walk matrix of the extended Dynkin graph

Let $\tilde{D}_n$ be the extended Dynkin graph with $n+1$ vertices, let $W(\tilde{D}_n)$ be its walk matrix, and set

$$r=\left\lfloor\frac{n}{2}\right\rfloor.$$

Smith normal form conjecture. The Smith normal form of $W(\tilde{D}_n)$ is

$$\begin{cases} \operatorname{diag}(\underbrace{1,\ldots,1}_{r-1},n-1,0,\ldots,0), & \text{if $n$ is even},\\ \operatorname{diag}\left(\underbrace{1,\ldots,1}_{r-1},\frac{n-1}{2},0,\ldots,0\right), & \text{if $n$ is odd}. \end{cases}$$

This conjecture gives the remaining Smith normal form data after the paper determines the rank of the walk matrix, and its verification would complete the proposed next step for the extended Dynkin graphs $\tilde{D}_n$.