Completion conjecture for Foulis–Randall products

Let $H_1,\ldots,H_n$ be contextuality scenarios, let $\overline{H_1\otimes\ldots\otimes H_n}$ denote the completion of their $n$-fold product scenario, let $\bar H_i$ denote the completion of $H_i$, and let ${}^{\max}\otimes_{i=1}^n H_i$ denote the maximal tensor product. Completion conjecture. One has

$$\overline{H_1\otimes\ldots\otimes H_n} = {}^{\max}\otimes_{i=1}^n \bar{H}_i.$$

This conjecture proposes that the completion of the product can be computed directly from the completions of the individual scenarios. It follows a result showing that the relevant Foulis–Randall products are associative up to observational equivalence, but the asserted formula itself is left as a conjecture.