The existence and uniqueness conjecture for the skew Brownian SDE flow

Let $\bm{\mathscr Z}_{\rho,q}^{(u)}$ be the solution flow of the skew Brownian SDEs driven by the correlated Brownian pair associated with parameters $\rho\in[-1,1]$ and $q\in[0,1]$.

Existence and uniqueness conjecture. Existence and pathwise uniqueness for the solutions of these SDEs hold for every

$$\rho\in[-1,1),\qquad q\in[0,1].$$

The case $\rho=-1$ reduces to the Harrison–Shepp skew Brownian equation, whose solutions are pathwise unique. The conjecture extends this well-understood extremal case to the full range $\rho<1$.