I thank the authors for putting all the relevant data in the supplements! I used Table S1, read into R, to ask whether there might be any interaction between neonics and "varokt" = the varroa levels in October. I used a simpler analysis method than the authors just to speed things up. I found that the best model using lm (by BIC criterion) is:
> summary(a5<-lm(loss~ pc2_2012 +varokt*acetamiprid ,data=r))
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.60363 0.26793 2.253 0.028426 *
pc2_2012 0.27258 0.11587 2.352 0.022402 *
varokt -0.17231 0.05149 -3.346 0.001511 **
acetamiprid -0.57269 0.24532 -2.334 0.023396 *
varokt:acetamiprid 0.19504 0.04987 3.911 0.000263 ***

Multiple R-squared: 0.4846, Adjusted R-squared: 0.4457
F-statistic: 12.46 on 4 and 53 DF, p-value: 3.263e-07

That is, even without doing the glm using a binomial link function, a simple lm of loss (0/1) has 48.5% of variance explained by the random postal code effect, presence/absence of acetamiprid, varokt, and most strongly, the interaction of varokt and acetamiprid. Other factors were not significant when added to this model.

This suggests that the combined effects of neonics and high mite loads are worse than either separately, that is, there is an interaction.

It would be interesting to see if this analysis holds with the more sophisticated glm modeling the authors used.