Equivariant genus-zero correspondence conjecture for the local projective line

Let $X=\mathcal{O}_{\mathbb{P}^{1}}(l_1,l_2,l_3)$ with $l_1+l_2+l_3=-2$, and let $\mathrm{GW}_{0,2}(X)$ and $P_{1,d[\mathbb{P}^{1}]}(X)$ denote the equivariant invariants defined in the local-curve setup. Equivariant local-curve genus-zero correspondence conjecture.

$$\mathrm{GW}_{0,2}(X)=P_{1,2[\mathbb{P}^{1}]}(X)+\frac{1}{8}P_{1,[\mathbb{P}^{1}]}(X).$$

This is the degree-two equivariant analogue of the stable-pair genus-zero correspondence, in the case where $P_{0,[\mathbb{P}^{1}]}(X)=0$; it is supported by the local computations in the paper but remains conjectural in the stated generality.