The XYZ Theorem of Shepp [1] states that for a given poset $P$. Consider the probability space of all the linear extensions of $P$, where each possible extension is equally likely. Then for any three elements $x, y, z \in P$, we have
$$\mathbb{P}[x \le y \textrm{ and } x \le z] \ge \mathbb{P}[x \le y] \cdot \mathbb{P}[x \le z].$$

Shepp's proof uses FKG inequality on a distributive lattice on $[m]^n$, in which $x \le y$ iff $x_1>y_1$ and $x_i-x_1 \le y_i-y_1$ for $i \ge 2$, and the meet and join are defined as:
$$(x \wedge y)_i=\min\{x_i-x_1, y_i-y_1\}+\max\{x_1, y_1\},$$
$$(x \vee y)_i=\max\{x_i-x_1, y_i-y_1\}+\min\{x_1, y_1\}.$$

I can verify that this gives a distributive lattice and thus FKG applies. But what is the motivation behind it, i.e. what makes Shepp believe that this is even a distributive poset?

[1] L.A. Shepp, _The XYZ conjecture and the FKG inequality_, Ann. Probab. **10** (1982) 824--827, doi:[10.1214/aop/1176993791](https://doi.org/10.1214/aop/1176993791)