The general kernel formula for higher-depth mock modular forms

Let $\mathcal{G}$ be the instanton generating function, expanded in terms of the completed generating functions $\widehat h_{p_i,\mu_i}$ and theta series $\vartheta_{\boldsymbol p,\boldsymbol \mu}$. Let $\Phi^{\mathcal E}_v$ and $\widetilde\Phi^{\mathcal E}_{v_0}$ denote the tree kernels, and let $\mathds{T}_n^{\rm S}$ be the relevant set of rooted trees. General kernel conjecture. The instanton generating function has the theta-series decomposition

$$\mathcal{G}=\sum_{n=1}^\infty\frac{2^{-n/2}}{\pi\sqrt{2\tau_2}}\left[\prod_{i=1}^{n}\sum_{p_i,\mu_i}\sigma_{p_i}\widehat h_{p_i,\mu_i}\right]e^{-S^{\rm cl}_p}\vartheta_{\boldsymbol p,\boldsymbol \mu}(\widehat\Phi^{\rm tot}_n,n-2),$$

where

$$\widehat\Phi^{\rm tot}_n=\Phi^{\scriptscriptstyle\,\int}_1\,\rm Sym\left\{\sum_{T\in\mathds{T}_n^{\rm S}}(-1)^{n_T-1}\left(\widetilde\Phi^{\,\mathcal E}_{v_0}-\Phi^{\,\mathcal E}_{v_0}\right)\prod_{v\in V_T\setminus\{v_0\}}\Phi^{\,\mathcal E}_v\right\}.$$

This conjecture gives a general formula for the kernels governing the theta-series decomposition, extending the cases established analytically through $n=4$ and explicitly discussed in the paper.