They also controlled for "age, sex, annual household income, economic status (eg, employed, retired, or unemployed), highest educational qualification, and couple status".
Taken together those are likely to capture socio-economic status in all but a few edge cases. With a very large study size, and a huge result (30% difference) it's unlikely that it can be explained by socioeconomic status.
If this were a small study or found a small result, then edge cases like lower income millennials with no degree who live in inherited houses in wealthy areas with trees might skew the results.
If you disagree with the studies methodology fine, but I'm satisfied that authors have done their due diligence with respect to socio-economic status.
Taken together those are likely to capture socio-economic status in all but a few edge cases. With a very large study size, and a huge result (30% difference) it's unlikely that it can be explained by socioeconomic status.
The study discusses this by the way.