Bounded-probe conjecture for the probing simple random walk

Consider a probing simple random walk (PSRW) $\{K_t\}$ on growing domains $\{\mathbb{D}_t\}$ in $\mathbb{Z}^d$, $d\ge2$, starting at $K_0=0$ with $\mathbb{D}_0=\{0\}$. At time $t$, it sends an $\mathcal{F}_t$-adapted number $m(t)$ of probes, each adding one site to the domain before the walk moves. In the setting of part (b) of Proposition general_soldier, each probe location is chosen according to the hitting measure of $\mathbb{D}_t^c$ for simple random walk started at $K_t$. Bounded-probe conjecture. There exist an $\mathcal{F}_t$-adapted process $m(t)$, uniformly bounded above by a non-random integer $\lambda_d$, and a PSRW with an almost surely transient sample path when $d\ge3$, and an almost surely recurrent sample path with $m(t)\ge1$ when $d=2$. If this conjecture is valid, it remains to determine whether it holds with $\lambda_d=1$ and whether the same conclusion holds for constant $m(t)=\lambda_d$, eliminating all control from the PSRW.