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The distance is strictly greater than $2$ --- as is true for any compact, nonhomeomorphic $K_1$ and $K_2$ when $K_1$ is totally disconnected (Cohen-Chu, Stud. Math. 1995, p.6). The question of an upper bound is more puzzling. See comments above by @YemonChoi.

The distance is strictly greater than $2$ --- as is true for any compact, nonhomeomorphic $K_1$ and $K_2$ when $K_1$ is totally disconnected (Cohen-Chu, Stud. Math. 1995, p.6). The question of an upper bound is more puzzling. @HannesThiel comment 6/13/2016 suggests distance $\le 16$.

The distance is strictly greater than $2$ --- as is true for any compact, nonhomeomorphic $K_1$ and $K_2$ when $K_1$ is totally disconnected (Cohen-Chu, Stud. Math. 1995, p.6). Comment above by @YemonChoi 6/9/2016 shows the distance is $\le 4$.

The distance is strictly greater than $2$ --- as is true for any compact, nonhomeomorphic $K_1$ and $K_2$ when $K_1$ is totally disconnected (Cohen-Chu, Stud. Math. 1995, p.6). The question of an upper bound is more puzzling. See comments above by @YemonChoi.

The distance is strictly greater than $2$ --- as is true for any compact, nonhomeomorphic $K_1$ and $K_2$ when $K_1$ is totally disconnected (Cohen-Chu, Stud. Math. 1995, p.6). More puzzling is a specific upper bound. None seems to have yet been offered here? None published? Is the distance greater than, say, $1000$?

The distance is strictly greater than $2$ --- as is true for any compact, nonhomeomorphic $K_1$ and $K_2$ when $K_1$ is totally disconnected (Cohen-Chu, Stud. Math. 1995, p.6). Comment above by @YemonChoi 6/9/2016 shows the distance is $\le 4$.