Stationary-initial-condition Brownian-bridge expansion conjecture

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

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

Here $b$ is a standard Brownian bridge whose average is represented by $\langle\cdot\rangle_b$, and $\max$ denotes the maximum over the bridge parameter. Let $\theta_{\mathrm{stat}}(s)$ be the explicit stationary coefficient defined in the source. Stationary-initial-condition Brownian-bridge expansion conjecture.

$$\left\langle\theta_{\mathrm{bb},n}(s;b)\right\rangle_b=\theta_{\mathrm{stat}}(s)+\mathcal{O}(s^{n+1}).$$

This conjecture gives the perturbative expansion for the stationary initial condition. It is motivated by exact results and remains without a direct probabilistic proof in the source.