Minimal total Markov basis size conjecture for contingency-table matrices

Let $M$ be the matrix in the equation defining the space of tables, and let $A$, $B$, and $C$ denote the index sets used to specify the table and its conditionals. A total Markov basis size conjecture asserts that a minimal Markov basis of $M$ contains exactly

$$|B|-1+(|C|-1)\times|B|\times|A|$$

elements. This would follow if the preceding conjecture on the Markov basis of the coefficient matrix were true. The paper provides supporting examples, but the asserted size remains unresolved there.