Random walk on thin ice?

Possibly more of an extended comment than an answer, but a classic paper is Lehman and Weiss [1] In this paper it is proved that, for a restricted (where "restricted" refers to the thin ice problem in your language) walk on the 2D lattice,

$(1)$ The probability the random walker survives $n$ steps without being trapped decreases exponentially in $n$.

$(2)$ The random walker is trapped at some point almost surely (which follows from $(1)$.)

$(3)$ The expected number of steps before getting trapped is finite (also follows from $(1)$.)

They remark that $(1)$ holds for higher dimensional lattices with arbitrary probabilities. The argument is essentially that the walker always has a non-zero probability of being trapped on its first visit to a given hypercube in the lattice. In 2D, this looks like a 'G' shape. I don't see why this wouldn't hold if $k > 1$. Calculating the expected number of steps before getting trapped is completely beyond me though.

[1] Lehman, R. Sherman, and George H. Weiss. “A Study of the Restricted Random Walk.” Journal of the Society for Industrial and Applied Mathematics, vol. 6, no. 3, 1958, pp. 257–278. JSTOR, www.jstor.org/stable/2098696.