Writing $\log|z|=\frac12\log(x^2+y^2)$ gives Laplacian zero away from the origin, so I is true. A global harmonic conjugate would be a single-valued argument function, which cannot exist around a loop enclosing zero: its change around the loop is $2\pi$. Thus only local conjugates exist on suitable slit domains. II is false, giving B.
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