General-term conjecture for the short-time expansion of the KPZ generating function

Let $g$ be the function used to define $Q_{t,\beta}(\sigma)$, let $\mathfrak{L}_i$ denote the corresponding coefficients, and let $\mathbf{n}=(n_1,\ldots,n_{r+2-2q})$. If the distinct values among the $n_j$ have multiplicities $m_1,\ldots,m_k$, define the symmetry factor $S(\mathbf{n})=m_1!\cdots m_k!$. For $\sigma<1$, the general-term conjecture. The short-time expansion of $\log Q_{t,\beta}(\sigma)$ has the form

$$\begin{aligned} \log Q_{t,\beta}(\sigma)={}&\frac{1}{\pi \sqrt{t}}\int_{0}^{+\infty}\mathrm{d}x\sqrt{x}\,g(\sigma e^{-x})+\frac{1}{4\pi^2}\int_0^\sigma\mathrm{d}u\,u\left[\partial_u\int_{-\infty}^{+\infty}\mathrm{d}p\,g(u e^{-p^2})\right]^2\\\\ &+\sum_{r=1}^{+\infty}\sum_{q=0}^{\frac{r}{2}+1}\frac{2^{r+1-3q}}{(2q+1)!}t^{r/2}\sum_{\substack{r+2\geq n_1\geq\ldots\geq n_{r+2-2q}\geq0\\\ _1+\ldots+n_{r+2-2q}=r-1+q}}\frac{c_{r,q}(\mathbf{n})}{S(\mathbf{n})}\prod_{j=1}^{r+2-2q}\frac{\mathfrak{L}_{n_j+1}}{n_j!}. \end{aligned}$$

Here the coefficients $c_{r,q}(\mathbf{n})$ are positive integers. The conjecture is motivated by the explicitly computed expansion through order $t^3$, but precise analytical expressions for arbitrary high orders were not obtained, so the general form remains open.