This is a more generalized strengthening of this previous question. A rigorous proof has been provided by Eric to that question. However, I believe there to be a more general claim.
Consider unbiased 2-d random walks $S_n$ where the $n$-th step has (non-random) length $c_i > 0$, with $\sum_{i=0}^{\infty} c_i = \infty$ and $\sum_{i=0}^{\infty} c_i^2 < \infty$ . When we say that a random walk crosses a number $a$, we mean that $a$ is sandwiched by $S_{n-1} $ and $S_n$ for some $n$ .
Then I conjecture that for any such random walk $S_n$ and any number $a$, the expected number of times that $a$ is crossed by $S_n$ is $\infty$
My justification:
The variance $S_n - S_m $ is bounded for every $n,m$. Thus the difference between the maximum and minimum of $S$ has finite expectation. Also the random walk will converge almost surely. But the total amount of distance traversed is infinite. Let $ Y_n$ be the amount of times that $S_n$ is crossed between $S_{m-1} $ and $S_m$. Intuitively, the expectation $E(Y_n)$ goes to $\infty$ when $n$ goes to $\infty$. The expected number of future returns / crosses becomes infinite in the tail of the walk. But on the other hand, for any $a$, $N>0$ and $\delta > 0$ , we can find some $a^*$ and $m>N$ , with $|a-a^*| < \delta $ and $P(S_m = a^*) > 0 $