The biased-sign construction conjecture for the Tanaka flow

Let $\bm e$ be a one-dimensional Brownian excursion, and let $\bm{\mathscr Z}_{1,1-p}^{(u)}$ denote the solution flow of the displayed Tanaka-type SDE, started at time $u$. The construction analogous to the uniform-sign construction uses signs $\bm s(\ell)\in\{\oplus,\ominus\}$ indexed by local minima $\ell$ of $\bm e$.

Biased-sign construction conjecture. For $\rho=1$ and every $p\in[0,1]$, the solutions $\bm{\mathscr Z}_{1,1-p}^{(u)}$ can be constructed using independent biased signs satisfying

$$\mathbb{P}(\bm s(\ell)=\oplus)=p.$$

For $p=1/2$, this reduces to the coalescing flow associated with the Tanaka SDE. The conjecture supplies the analogous explicit construction for all biases.