Mutual absolute continuity of the Slepian and shifted Brownian zero sets

Let $S$ be the Slepian process, let $B$ be standard Brownian motion, and let $\xi\sim\mathcal{N}(0,1)$ be independent of $B$. For $t\geq 0$, consider the zero sets

$$\{u\in[0,t]:S_u=0\}\quad\text{and}\quad\{u\in[0,t]:\xi+B_u=0\}.$$

Zero-set absolute-continuity conjecture. For every $t\geq 0$, the Slepian zero set on $[0,t]$ is mutually absolutely continuous with respect to the zero set of Brownian motion started at $\xi$:

$$\{u\in[0,t]:S_u=0\}\quad\text{and}\quad\{u\in[0,t]:\xi+B_u=0\}\text{ are mutually absolutely continuous.}$$

The paper establishes the corresponding local structure on the unit interval through a path decomposition and explains that the Gaussian-measure absolute continuity used there fails beyond the unit interval. The conjecture proposes that mutual absolute continuity of the zero sets nevertheless holds on every finite interval, including intervals longer than $1$.