The determinant formula for the reduced walk matrix of the Dynkin graph $D_n$

Let $D_n$ ($n\ge 4$) be the Dynkin graph, let $W(D_n)$ be its walk matrix, and let $\hat{W}(D_n)$ be the $(n-1)\times(n-1)$ matrix obtained from $W(D_n)$ by removing the first row and the last column. Determinant conjecture. The determinant satisfies

$$\det \hat{W}(D_n)= \begin{cases} \pm 2^{\lfloor\frac{n}{2}\rfloor-1} & \text{if }4\nmid n,\\ 0 & \text{if }4\mid n. \end{cases}$$

The paper presents this formula as a conjecture and proves it, thereby confirming a recent conjecture of Wang, Liu, and Wang concerning the walk matrix of $D_n$.