Conjectured free-boundary solution for the EWF in Case IID

Let $B=(B_1,B_2)$ be the two-dimensional Brownian motion on a filtered probability space $(\mathnormal{\Omega},\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P})$, and consider the EWF with $hat h$ as in the linear-programming value and parameters $(\mu_1,\mu_2,\mu_3)$ and $(c_1,c_2,c_3)$ as in Case IID. For a real number $x$, write $[x]^- = \max\{-x,0\}$. The EWF free-boundary conjecture. There exist functions $\Psi_i:\mathbb{R}_+\to\mathbb{R}_+$, $i=1,2$, that are Lipschitz and strictly increasing, with $\Psi_i(x)\to\infty$ as $x\to\infty$, such that there is a unique pair of $\{\mathcal{F}_t\}$-adapted continuous processes $W_1^*,W_2^*$ with values in $\mathbb{R}_+$ satisfying

$$W_1^*(t)=B_1(t)+\sup_{0\le s\le t}\left[B_1(s)-\Psi_1(W_2^*(s))\right]^-,$$ $$W_2^*(t)=B_2(t)+\sup_{0\le s\le t}\left[B_2(s)-\Psi_2(W_1^*(s))\right]^-.$$

Moreover, the processes

$$I_1^*(t)=\sup_{0\le s\le t}\left[B_1(s)-\Psi_1(W_2^*(s))\right]^-,\qquad I_2^*(t)=\sup_{0\le s\le t}\left[B_2(s)-\Psi_2(W_1^*(s))\right]^-$$

form an optimal control for the EWF. The conjecture concerns the currently unavailable EWF and Brownian control problem in regime IID and would be a key step toward constructing asymptotically optimal control policies there; Case IID is described as a challenging open problem.