Toric Residue Mirror Conjecture for the first Hirzebruch surface

Let ${\mathbb P}_{\Sigma}$ be the toric surface associated with the triangulation ${\mathcal T}$, and let $D_1,\ldots,D_4$ be its torus-invariant divisors corresponding to $v_1,\ldots,v_4$. Fix a homogeneous polynomial $P(x_1,\ldots,x_4)\in\mathbb Q[x_1,\ldots,x_4]$ of degree two. For $\beta=(b_1,\ldots,b_4)\in K_{\rm eff}({\mathbb P}_{\Sigma})$, define

$$\Phi_\beta=([D_1]+\cdots+[D_4])^{b_1+\cdots+b_4}\prod_{i:b_i<0}[D_i]^{-b_i-1},\qquad a^\beta=a_1^{b_1}\cdots a_4^{b_4}.$$

Toric Residue Mirror Conjecture for the first Hirzebruch surface. The series expansion at the vertex $v_{\mathcal T}\in\operatorname{Sec}(A)$ of the toric residue

$$R_P(a_1,\ldots,a_4)=\operatorname{Res}_f\bigl(t_0^2P(a_1t^{v_1},\ldots,a_4t^{v_1})\bigr)$$

coincides with the generating function

$$I_P(a_1,\ldots,a_4)=\sum_{\beta\in K_{\rm eff}({\mathbb P}_{\Sigma})}\langle P([D_1],\ldots,[D_4])\Phi_\beta\rangle_\beta a^\beta.$$

This is presented as a reformulation of the general Toric Residue Mirror Conjecture in the case ${\mathbb P}={\mathbb F}_1$; the supplied text gives no resolution status.