Does having women in top positions help close the gender pay gap?

This fascinating research from the US suggests the answer might be "no". They find no difference in the earnings penalty experienced by mothers whether the three top earners are all male or all female... https://twitter.com/dhsandler/status/1354878305749651456
... and this is particularly important since in countries like the US, the motherhood penalty explains a *lot* of the earnings and career progression gaps for otherwise like-for-like men and women in the labor force.
Note this result could be compatible with a number of possibilities.

(1) may suggest that the motherhood pay penalty is more to do with structural aspects of workplace culture that are *beyond* what an individual firm's management can impact? (e.g. industry norms re work hours)
(2) may suggest that the women who make it to the top are as (un)likely as men to implement workplace policies that would reduce motherhood pay penalty?
Giving rise to possibility (4) - that the biggest factor impacting the child earnings penalty may be decisions within the household about allocation of domestic labor, childcare, and workplace policies may be a smaller factor
Very interested in other thoughts!
Another interesting paper: https://twitter.com/arpitrage/status/1354906447407177731
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