Reflecting-boundary strategy for unbounded harvesting

Consider the population process with unbounded harvesting governed by

$$d\tilde X(t)=\tilde X(t)(\mu-\kappa\tilde X(t))\,dt+\sigma\tilde X(t)\,dB(t)-dZ_t,$$

where $(Z_t)_{t\geq0}$ is a non-negative, increasing, right-continuous process adapted to $(\mathcal{F}_t)_{t\geq0}$. Assume that the population survives without harvesting, namely $\mu-\frac{\sigma^2}{2}>0$. Reflecting-boundary conjecture. The optimal extraction strategy $(Z_t^*)_{t\geq0}$ has the form

$$Z_t^*(x)=\begin{cases}(x-x^*)^+ & \text{if }t=0,\\ L(t,x^*) & \text{if }t>0, \end{cases}$$

for some $x^*\in(0,\infty)$, where $L(t,x^*)$ is the local time at $x^*$ of the process $\tilde X$. This conjecture extends the bounded-rate bang-bang result to unbounded harvesting: after an initial removal above the threshold, harvesting is represented by local time at a reflecting boundary. The source presents this as a conjectured optimal solution and does not provide a proof.