The curve analogue of Nekrasov's conjecture for the resolved conifold

Let $X=\operatorname{Tot}_{\mathbb P^1}(\mathcal O(-1)\oplus\mathcal O(-1)\oplus\mathcal O)$, and let $\mathcal Z_X(y,q,Q)=\sum_{n,d}P_{n,d[\mathbb P^1]}(\mathcal O,y)q^nQ^d$ be its stable-pairs generating series. Let $t_4^{-1}$ be the torus weight of $\mathcal O$ over $\mathbb P^1$, and let $[x]=x^{1/2}-x^{-1/2}$. The curve analogue of Nekrasov's conjecture. There exist unique choices of signs such that

$$\mathcal Z_X(y,q,Q)=\operatorname{Exp}\left(\frac{Q[y]}{[t_4][y^{\frac12}q][y^{\frac12}q^{-1}]}\right).$$

Here the plethystic exponential is applied to all variables. The formula is presented as a new conjectural expression for stable-pairs invariants of the resolved conifold, analogous to the empty fourfold vertex formula.