Huang–Jiang's conjectural evaluation for the even torus-link singularity

Let $R^{(2,2k)}$ be the germ of the plane curve singularity $y(y-x^k)=0$, and let $\widehat{N\!Z}_{R^{(2,2k)}}(t)$ denote its motivic Cohen–Lenstra zeta function. Here $k$ is a positive integer. Huang–Jiang's conjecture. For any positive integer $k$,

$$\widehat{N\!Z}_{R^{(2,2k)}}(-1)=\prod_{j\ge 1}\frac{(1-\mathbb{L}^{-2j})(1-\mathbb{L}^{-(k+1)j})^2}{(1-\mathbb{L}^{-j})^2(1-\mathbb{L}^{-(2k+2)j})}.$$

This conjecture gives a product evaluation of the motivic zeta function at $t=-1$; the corresponding specialization at $t=1$ is known to equal $1$, while the displayed evaluation was proposed by Huang and Jiang and remains conjectural in the supplied source.