Flat-initial-condition Brownian-bridge expansion conjecture

Let $n$ be a non-negative integer. For independent standard Brownian bridges $b_0,\ldots,b_{2n}$, define

$$\theta_{\mathrm{bb},n}(s;0)=\frac{\left\langle\exp\left(-s\sum_{j=1}^n\max[b_j-b_{j-1}]\right)\right\rangle_{b_0,\ldots,b_n}\left\langle\exp\left(-s\max[b_1]-s\sum_{j=2}^n\max[b_j-b_{j-1}]\right)\right\rangle_{b_1,\ldots,b_n}}{\left\langle\exp\left(-s\sum_{j=1}^{2n}\max[b_j-b_{j-1}]\right)\right\rangle_{b_0,\ldots,b_{2n}}}.$$

Here $\max$ denotes the maximum over the bridge parameter. Let $\chi(v)=-\operatorname{Li}_{5/2}(-\exp(v))/\sqrt{2\pi}$, let $\nu(s)$ solve $\chi'(\nu(s))=s$, and define $\theta_{\mathrm{flat}}(s)$ by the explicit flat-initial-condition formula in the source. Flat-initial-condition Brownian-bridge expansion conjecture.

$$\theta_{\mathrm{bb},n}(s;0)=\theta_{\mathrm{flat}}(s)+\mathcal{O}(s^{n+1}).$$

This conjecture gives the perturbative expansion of the flat-initial-condition coefficient in terms of finitely many Brownian bridges. The source reports agreement with exact results and presents it as an unproved conjecture.