Brownian-bridge perturbative expansion conjecture for TASEP cumulants

Let $n$ be a non-negative integer and let $h_0$ be an arbitrary sufficiently regular continuous function satisfying

$$h_0(0)=h_0(1)=0.$$

For independent standard Brownian bridges $b_1,\ldots,b_n$, define

$$f_{\mathrm{bb},n}(s;h_0)=-\frac{\left\langle b_n'(0^+)b_n'(1^-)\exp\left(-s\max[b_1-h_0]-s\sum_{j=2}^n\max[b_j-b_{j-1}]\right)\right\rangle_{b_1,\ldots,b_n}}{\left\langle\exp\left(-s\max[b_1-h_0]-s\sum_{j=2}^n\max[b_j-b_{j-1}]\right)\right\rangle_{b_1,\ldots,b_n}}.$$

Here $\max$ denotes the maximum over the bridge parameter, and define $\chi(v)=-\operatorname{Li}_{5/2}(-\exp(v))/\sqrt{2\pi}$; let $\nu(s)$ solve $\chi'(\nu(s))=s$. Brownian-bridge perturbative expansion conjecture.

$$f_{\mathrm{bb},n}(s;h_0)=\frac{s^2}{3}+\frac{\chi(\nu(s))}{s}+\mathcal{O}(s^n).$$

In particular, this perturbative expansion is independent of $h_0$ through order $s^{n-1}$. The conjecture arises from exact Bethe-ansatz formulas and corresponding Brownian-bridge representations; direct probabilistic proofs were unknown in the source, while low-order cases had numerical support.