Conditional geodesic fluctuation conjecture

Let $\pi_*$ be the unique geodesic from $(1,1)$ to $(aN,bN)$, linearly interpolated and written in the basis $\mathbf v_1=(a,b)$ and

$$\mathbf v_2=\left(\frac{a(\ell-a+b)\sigma}{\ell\sqrt D},-\frac{b(\ell+a-b)\sigma}{\ell\sqrt D}\right),$$

as $\pi_* =\{\tau\mathbf v_1+\pi^*(\tau)\mathbf v_2\}_{\tau\in[0,N]}$, where $\pi^*(0)=\pi^*(N)=0$. Let $\mathbb B^{\mathrm{br}}_1$ and $\mathbb B^{\mathrm{br}}_2$ be the correlated Brownian bridges from the diagonal fluctuation theorem.

Conditional geodesic fluctuation conjecture. Conditional on $\mathcal L(aN,bN)=\ell N$,

$$\operatorname{Law}\left(\left(\frac{\pi^*(tN)}{N^{1/2}},\frac{\mathcal L(tN\mathbf v_1+\pi^*(tN)\mathbf v_2)-t\ell N}{\sigma N^{1/2}}\right)_{t\in(0,1)}\,\middle|\,\mathcal L(aN,bN)=\ell N\right) \xrightarrow{\mathrm{f.d.d.}} \operatorname{Law}\left((\mathbb B^{\mathrm{br}}_2(t),\mathbb B^{\mathrm{br}}_1(t))_{t\in(0,1)}\right).$$

The conjecture identifies the joint scaling limit of the transverse geodesic displacement and the passage-time fluctuation along the conditioned geodesic. It is motivated by the diagonal fluctuation theorem and remains open.