The local-curve stable-pair generating-series conjecture

Let $X=\operatorname{Tot}_{\mathbb P^1}(\mathcal O\oplus\mathcal O(-1)\oplus\mathcal O(-1))$ and let $P_{n,d}(X;T)$ be the equivariant stable-pair invariant, where $d$ is the degree over the zero section and $q,y$ are formal variables. Let $\lambda_2$ be the equivariant parameter for the action on the first fibre $\mathcal O_{\mathbb P^1}$. Local-curve stable-pair generating-series conjecture. There exist choices of signs such that

$$\sum_{n,d\geqslant0}P_{n,d}(X;T)q^ny^d=\exp\left(\frac{qy}{\lambda_2}\right),$$

with $P_{0,0}(X;T)=1$. This is a conjectural closed formula for the stable-pair generating series of the local curve; its general proof is not supplied in the source.