The problem of distributed testing against independence with variable-length coding is considered when the \emph{average} and not the \emph{maximum} communication load is constrained as in previous works. The paper characterizes the optimum type-II error exponent of a single sensor single decision center system given a maximum type-I error probability when communication is either over a noise-free rate-$R$ link or over a noisy discrete memoryless channel (DMC) with stop-feedback. Specifically, let $ε$ denote the maximum allowed type-I error probability. Then the optimum exponent of the system with a rate-$R$ link under a constraint on the average communication load coincides with the optimum exponent of such a system with a rate $R/(1-ε)$ link under a maximum communication load constraint. A strong converse thus does not hold under an average communication load constraint. A similar observation holds also for testing against independence over DMCs. With variable-length coding and stop-feedback and under an average communication load constraint, the optimum type-II error exponent over a DMC of capacity $C$ equals the optimum exponent under fixed-length coding and a maximum communication load constraint when communication is over a DMC of capacity $C(1-ε)^{-1}$. In particular, under variable-length coding over a DMC with stop feedback a strong converse result does not hold and the optimum error exponent depends on the transition law of the DMC only through its capacity.