Sahi's conjecture for three increasing events

Let $\Omega$ be a product space, let $\mu$ be a product measure on $\Omega$, and let ${\mathcal A},{\mathcal B},{\mathcal C}\subseteq\Omega$ be increasing events. Sahi's conjecture. One has

$$2\mu({\mathcal A}{\mathcal B}{\mathcal C})-[\mu({\mathcal A}{\mathcal B})\mu({\mathcal C})+\mu({\mathcal A}{\mathcal C})\mu({\mathcal B})+\mu({\mathcal B}{\mathcal C})\mu({\mathcal A})]+\mu({\mathcal A})\mu({\mathcal B})\mu({\mathcal C})\geq 0.$$

This is proposed as the three-event extension of Harris' inequality; the paper states that the case $k=3$ has proved intractable and may capture the full difficulty of the general problem.