The general-potential extension of the conditioned Brownian-motion limit

Let $Q$ be a rate function, let $\theta>0$, and define

$$I^{(Q)}_T:=\int_0^T Q(W_s)\,\mathrm{d}s.$$

For a starting point $x$, write $\mathbb{P}_x$ for the law of Brownian motion started at $x$, and let $\mathcal{C}([0,\infty))$ denote the space of continuous paths. For a parameter $\beta$, let $\phi=\phi^{(\beta)}$ be an eigenfunction corresponding to the smallest eigenvalue $\lambda=\lambda^{(\beta)}$ of

$$\phi”(x)-\beta Q(x)\phi(x)=-\lambda\phi(x).$$

General-potential conditioning conjecture. As $T\to\infty$, the probability measures

$$\mathbb{P}_x\left((W_t)_{t\geq 0}\in\cdot\,\middle|\,I^{(Q)}_T\leq\theta T\right)$$

converge weakly on $\mathcal{C}([0,\infty))$ to the law of a diffusion process $(X_t)_{t\geq 0}$ satisfying

$$X_0=x,\qquad \mathrm{d}X_t=\mathrm{d}B_t+\frac{\phi'}{\phi}(X_t)\,\mathrm{d}t,\quad t\geq 0,$$

where $(B_t)_{t\geq 0}$ is a standard Brownian motion and the parameter $\beta=\beta(\theta)$ depends on $\theta$ in an implicit way. This is proposed as an extension of the paper's conditioned Brownian-motion result from the quadratic potential to more general rate functions $Q$; an analogous exponential estimate is stated to hold under fairly general assumptions, while the corresponding process-level convergence remains conjectural.