48 children were enrolled in the trial. Half of them were given the treatment and half the placebo, leaving 24 children in each group. Statistical significance testing is reported in the article and seems fairly robust, but this is too small a sample size to be fully confident in the results.
Of 24 children treated, apparently 20 of them were successfully treated.
The control group only saw one person's allergy diminish during the initial treatment period.
I'll freely admit I don't have a background in statistics or clinical trial design, but it seems to me with that significant of an effect you can be fairly confident in the results even at that trial size.
If we were talking 8% to 4% in 24-person groups, sure, one extra person having a positive result could be random chance. We're talking 82% to 4%. It seems pretty unlikely that 19 people all randomly has a positive result unrelated to the treatment.
How big do you think these trial groups need to be to confirm the effect of the treatment, if this isn't good enough?