Axis-minimization conjecture for monotonic path covering

For integers $L\ge N\ge1$, let $\mathcal{P}$ be any nearest-neighbor monotonic path in $\mathbb{Z}^d$ of length $N$, and let $\{X_n\}_{n\ge0}$ be a $d$-dimensional simple random walk starting at $0$. Define the straight coordinate-axis path

$$\overset{\rightarrow}{\mathcal{P}}=\big((0,0,\ldots,0),(1,0,\ldots,0),\ldots,(N-1,0,\ldots,0)\big).$$

Axis-minimization conjecture.

$$P\big({\rm Trace}(\mathcal{P})\subseteq {\rm Trace}(X_0,\ldots,X_L)\big)\ge P\big(\overset{\rightarrow}{\mathcal{P}}\subseteq {\rm Trace}(X_0,\ldots,X_L)\big).$$

Thus, among monotonic paths of length $N$, the covering probability is conjectured to be minimized by a straight path along a coordinate axis. The source presents this as an additional conjecture, with no resolution supplied.