Source? The abstract states definitively that there was statistically significant increase on the gluten pill.

From the study:

No significant difference (p=0.242) was found in the overall score between sequences, i.e. (gluten→placebo) and (placebo→gluten).

And then:

9 out of 59 would suggest that this was not the case.

Possibly, but the study goes on to say:

Our attention was conversely focused on the 9 patients (15%) localized in the lower right region of the diagram, that is on those patients strongly suspected to be true gluten-sensitive according to their high positive gap between gluten and placebo scores. Thus, we plotted patients on the basis of their delta overall scores, which ranged from - 103 to +156, with a mean (SD) value of 12.2 (50.4) (Figure 3B, dashed line). According to our definition (mean +2SD), the cut-off level of the delta overall score was +113 (dotted line). Only three patients had a delta overall score > 113, and thus were identified as true gluten-sensitive.

So, no, only 3 out of 59 would support that it would appear. Of course, some dropped out too early to complete the trial due to symptoms so that might affect things. Also it's based on self-reported gluten sensitivity, which could also muddy things.

Statistically significant random chance?

Yes, that happens sometimes especially with smaller groups or groups with less statistical power. Certainly possible, though the trick is to narrow down how likely it is and replicate to remove more doubt.

Well we know that a small percentage of people who think they are sensitive to gluten actually have celiac and celiac is real, so I don't know why this author appears so certain that gluten sensitivity probably doesn't exist. This just reads as someone who really doesn't want it to exist, rather than a robust analysis of the study.

Well, this is the author's own words in their conclusion:

In conclusion, in the present trial most patients showed approximately equal degrees of overall symptoms under either gluten or placebo, although overall symptoms were significantly worsened by gluten in comparison to placebo. As regards the identification of the true gluten-sensitive patients, it should be cautiously interpreted due to the lack of a control group of non-gluten-sensitive subjects, and it does not represent a crucial evidence in favor of the existence of this new syndrome. We cannot exclude the possibility that these patients merely had increased visceral or extraintestinal hypersensitivity to gluten, nor that higher gluten doses could have selected a larger cohort of sensitive patients.

What's extremely confusing and easy to misinterpretation is that the degree to which the patients experienced symptoms was not statistically different. The overall symptoms were. So, if my understanding is correct, both groups experienced the same degree of negative symptoms but those on the gluten regiment experienced broader range of negative symptoms over placebo.

The author's describe this further:

Actually, we found that the overall symptom score was significantly higher under gluten in comparison to placebo. However, when we examined the individual patients’ overall scores we found that only a minority of the participants experienced a real worsening of symptoms under gluten. While it is possible that the global evaluation of the symptoms may in some way have attenuated the effect played by gluten on predominant symptoms, i.e. abdominal bloating and pain, we do acknowledge that the relevance of NCGS should be reappraised. This view is also supported by the evidence that in the vast majority of patients the clinical weight of gluten-dependent symptoms is irrelevant in the light of the comparable degree of symptoms experienced with placebo. If we look at the distribution of delta overall scores (gluten minus placebo), it is not surprising to note that a fair number of patients are victims of the nocebo effect, which was extensively proved through double-blind, placebo-controlled trials.31,32

So, more research needs to be done. Results could easily be explained by nocebo, actual effects of drastic diet changes with gluten, or Non-Celiac's Gluten Sensitivity.