Equivariant Joyce–Song stable-pair formula for the local curve $\mathcal{O}_{\mathbb{P}^{1}}(-1,-1,0)$

Let $X=\mathcal O_{\mathbb P^1}(-1,-1,0)$ and let $T$ be the Calabi–Yau torus, with equivariant parameter $\lambda_3$. Let $P^{\mathrm{JS}}_{n,d}$ denote the $T$-equivariant Joyce–Song stable-pair invariant. Local-curve Joyce–Song conjecture. The invariants satisfy

$$P^{\mathrm{JS}}_{n,d}=\begin{cases} \dfrac{1}{d!(\lambda_3)^d},& n=d\geqslant0,\\ 0,&\text{otherwise}. \end{cases}$$

This is presented as a direct local analogue of the main wall-crossing conjecture. The source calls it expected and does not provide a proof in the supplied span.