 Origin capture by a sufficiently sleepy planar random walk

Suppose two independent planar simple symmetric random walks are compared while one stays at its $j$th position throughout a deterministic time block. If each block has an unconditional probability bounded below by a positive constant that the other walk visits the origin, simultaneous origin visits occur infinitely often almost surely. Markov-chain recurrence gives infinitely many origin blocks for the sleepy walk. The block bound yields positive probability of infinitely many coincidences; eventual coupling of planar symmetric random walks then makes this future event have probability zero or one. Conditional block bounds or independence of capture attempts are not required.