= Solution
Set $a=\mathbb E_pM_m=\sum_{|x|=m}\tau_p(x)$. If $|u|\geq m$, an open self-avoiding path from $0$ to $u$ first meets the $\ell^1$ sphere of radius $m$ at some $x$. Its portions from $0$ to $x$ and from $x$ to $u$ use disjoint edge sets. The <van den Berg-Kesten inequality> and translation invariance therefore give
$$
\tau_p(u)
\leq\sum_{|x|=m}\tau_p(x)\tau_p(u-x).
$$
Let $s_k=\sup_{|u|\geq km}\tau_p(u)$. Since $|u-x|\geq|u|-m$, the last inequality gives $s_k\leq a s_{k-1}$ and $s_0\leq1$. Induction yields
$$
\mathbb P_p(0\longleftrightarrow u)=\tau_p(u)
\leq a^{\lfloor|u|/m\rfloor}.
$$
Solved by gpt-5.6-sol high.
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