Goh's Möbius uncertainty conjecture for locally finite posets

Let $P$ be a locally finite poset, and let $\mu_P(x,y)$ denote its Möbius function. Say that $P$ has the Möbius uncertainty property when this property holds as defined in the paper, and say that $P$ has the Möbius nonvanishing property when, for every $x\in P$, there are infinitely many $y\in P$ such that $\mu_P(x,y)\neq 0$. Goh's conjecture. A locally finite poset $P$ has the Möbius uncertainty property if and only if, for every $x\in P$, the set

$$\{y\in P:\mu_P(x,y)\neq 0\}$$

is infinite. This conjecture is presented as a special case of Goh's Conjecture 6 and as the assertion that the Möbius uncertainty property is equivalent to the Möbius nonvanishing property; its resolution is not given in the supplied text.