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They do mention it. They suggest that those types of ailments are increasing, but not at a pace that would explain these deaths.

"...the study clearly shows they are not the result of diseases such as lung cancer or diabetes, which are declining and increasing slowly, respectively."

They are examining trends in morbidity data. Presumably, those causes of death did not show the same rise that is shown in the causes of death that they are calling attention to.

That is, the paper would mention it if the population in question had seen an unusual rise in deaths from cardiovascular diseases and cancers.

But aren't all other segments of the population experiencing that as well?