Caravenna-Deuschel conjecture for integrated random bridges

Let $(X_i)$ be centered i.i.d. increments with law $\mu$ and finite variance, let $S_n$ and $A_n$ be the associated random walk and integrated walk, and write $\Omega_{N-1}^+=\{A_1\ge0,\ldots,A_{N-1}\ge0\}$. Caravenna-Deuschel conjecture. For every such increment law,

$${\mathbb{P}}\left[\Omega_{N-1}^+\mid A_N=A_{N+1}=0\right]\asymp N^{-1/2}.$$

This concerns entropic repulsion for the integrated random walk conditioned to return to the origin; it is stated as an open result in the source.