The multiterminal hypothesis testing problem with zero-rate constraint is considered. For this problem, an upper bound on the optimal error exponent is given by Shalaby and Papamarcou, provided that the positivity condition holds. Our contribution is to prove that Shalaby and Papamarcou's upper bound is valid under a weaker condition: (i) two remote observations have a common random variable in the sense of Gácks and Körner, and (ii) when the value of the common random variable is fixed, the conditional distribution of remaining random variables satisfies the positivity condition. Moreover, a generalization of the main result is also given.