Shifted $L_\infty$ master equation for spliced treed holomorphic discs

Let $X^{00}$ be the relevant open space, let $\pi^*\mathcal{O}_{an}$ be the pulled-back analytic-function sheaf, and let $C^*(X^{00},\pi^*\mathcal{O}_{an})$ be its Morse complex. For $m\geq 1$, consider operations

$$\ell_m:C^*(X^{00},\pi^*\mathcal{O}_{an})^{\otimes m}\to C^*(X^{00},\pi^*\mathcal{O}_{an})[3-2m]$$

defined by counts of spliced configurations with $m-1$ splicing edges and one infinite gradient flow line, with no disc components; $\ell_1$ is the Morse differential and $\ell_2$ is, up to sign, the bracket. Spliced-disc $L_\infty$ conjecture. These operations define a shifted $L_\infty$-structure, and weighted counts of rigid spliced treed holomorphic discs define $\mathfrak{m}_0^{spl}\in C^*(X^{00},\pi^*\mathcal{O}_{an})$ satisfying

$$\sum_{m\geq 1}\frac{1}{m!}\ell_m\bigl((\mathfrak{m}_0^{spl})^{\otimes m}\bigr)=\delta\mathfrak{m}_0^{spl}\pm\frac12\{\mathfrak{m}_0^{spl},\mathfrak{m}_0^{spl}\}+\frac16\ell_3(\mathfrak{m}_0^{spl},\mathfrak{m}_0^{spl},\mathfrak{m}_0^{spl})+\dots=0.$$

The equation is proposed as the algebraic summary of codimension-one boundary contributions from spliced configurations; the source does not establish the full shifted $L_\infty$ structure or master equation.