This study is a retrospective study, not a causal study – so they can only observe correlations. If, for instance, there is a genotype that gets LADA more often and also increases your liking of salt, it’s not the salt that causes LADA. There can be a co-factor at play.
However, there is evidence of a steady increase in T1 and LADA rates over the past 40-50 years. So many (most?) epidemiologists believe that there are environmental factors at play.
I would also suggest that, as @MaryPat suggests, we don’t KNOW that LADA is purely genetic, in the same manner as we are now discovering many genetic factors to T2D.