Well, the data indicate that this is not the case. For households in the bottom two quintiles of income, only 18(!)% of householders work full time in the bottom quintile, and 45% work full time in the next-lowest quintile.
source: https://www.aei.org/carpe-diem/explaining-income-inequality-...
At least over here (Poland, EU), people officially unemployed often work under the table, sometimes quite a lot, and they cannot afford the word of it to get to any public agency. Such people are unwilling to talk about it to researchers either.
(We recently had a public census, and it asked a bunch of sensitive questions around topics like employment. Despite the survey repeatedly giving assurances that this is all used for public statistics, and that data will be processed by trained personnel under strict confidentiality regulation, the people I know who are forced to skirt employment/tax rules here and there to survive, were not willing to reveal any indication of it.)
"Job finding office" literally asks you whether you're willing to work "under the table".
I've been asked this question when attempted to find my very first job
So legally they'd only show up as working 15-20-35 hours and in fact they'd work 60 hours a week.