Del Zotto–Lockhart's conjecture for rank-one instanton strings

Let $G$ be the global symmetry group of a rank-one $H_G$ theory, let $v$ and $m_G$ denote its fugacities, and let $Q_{\tau}$ be the elliptic-parameter fugacity. Write

$$L_{G}^{(1)}(v,m_G,Q_{\tau})=\sum_{i,j=0}^{\infty}b^G_{i,j}Q_{\tau}^iv^j.$$

Let $\mathbb{E}_{h_G^{(1)}}(v)$ denote the reduced one-string elliptic genus, set $h=h_G^\vee/6$, and define $u=v/Q_{\tau}^{1/4}$. Del Zotto–Lockhart's conjecture. There exists such a function $L_G^{(1)}$ satisfying: each $b^G_{i,j}$ is a sum of characters of irreducible representations of $G$ with integral coefficients; $L_G^{(1)}(v,m_G,0)$ is the Hilbert series of the reduced moduli space of one $G$-instanton, equivalently the Hall–Littlewood index of the $H_G^{(1)}$ theory; $L_G^{(1)}(q^{1/2},m_G,q^2)$ is the Schur index of the $H_G^{(1)}$ theory; and $\mathbb{E}_{h_G^{(1)}}(v)$ is generated from $L_G^{(1)}(v)$ by the formula

$$\mathbb{E}_{h_G^{(1)}}(v)=v^{2h-1}Q_{\tau}^{1/6}\sum_{n\geq0}Q_{\tau}^{2n}\bigg[u^{4h}L_G(Q_{\tau}^n v)-(-1)^{2h}u^{-4h}L_G(Q_{\tau}^{n+1/2}/v)+(1+(-1)^{2h})Q_{\tau}^{h+1/2}\left(u^2L_G(Q_{\tau}^{n+1/2}v)-u^{-2}L_G(Q_{\tau}^{n+1}/v)\right)+Q_{\tau}^2\left((-1)^{2h}u^{4(1-h)}L_G(Q_{\tau}^{n+1}v)-u^{-4(1-h)}L_G(Q_{\tau}^{n+3/2}/v)\right)\bigg].$$

This conjecture proposes a common structure relating one-string elliptic genera to the Hilbert series and supersymmetric indices of rank-one $H_G$ theories. Its resolution is not specified in the source.