Gini index fell everywhere in great depression, great recession, the 80s recession, and so on. Has this ever not been the case? Have any countries not seen inequality fall in recession?
What you suggest can easily be rationalised, if those on fixed incomes (such as pensioners) tend to be poorer while those with economically sensitive incomes (such as workers and shareholders) tend to be richer and see their incomes more adversely affected by recessions.
But I am not convinced it is true in every case. The UK recession in 1980 may be an example, looking at this graph from the Office for National Statistics and I would expect there to be examples for other countries too.