The key driver of this tight relation between inequality and r is the fact that the wealthiest households have much duration than other households, leading to the concentration of capital gains in the right tail of the wealth distribution when r declines.
By the way, while it is a widely held view that wealth inequality is high when r-g is large because of Piketty's book, this is not true among economists who understand finance or have just looked at the data.
As the graph below shows, there is a perfect time correlation between the level of wealth concentration and the inverse of r (1/r, represented here by the price of bonds).
The fact that the correlation between wealth inequality and r-g in the data is the opposite of what the Piketty story predicts was pointed early on in Acemoglu and Robinson (2015).
It's incredible that a story that is rejected by casual regressions/plotting the data is so popular.
Intuitively, falling interest rates have the direct mechanical effect of increasing the market value of assets, and in particular that of long-term assets such as start ups and stocks, which tends to be disproportionally owned by the rich. It also inflates the fair value of non-tradable assets such as Social Security benefits which disproportionnally accrue to the middle-class but are left out of these computations in most of the literature.