Nobody in the West will receive funding to do this study, unless they are doing it for a conservative think tank I suspect. What I can say is China has started doing the studies on it though.
The issue of whether abortion increases the risk of future mental health problems for women remains a debated topic, and there is a lack of evidence from large-scale studies conducted in China. This study aimed to investigate the potential ...
pmc.ncbi.nlm.nih.gov
"Multiple logistic regression analysis revealed that,
compared with women without any history of abortion, postmenopausal women who reported three or more abortions during their childbearing years were likely to have worse mental health conditions (OR [95% CI]: 1.37 [1.13, 1.67]). Additionally, women who reported a history of abortions for socio-economic reasons were also correlated with an increased risk of mental health issues after menopause (OR [95% CI]: 1.34 [1.08, 1.66])."
That is one study. There may be more. If I come across them, I will put them up.
Edit: Found another that does not seem to be from a bias source.
"Results
Rates of mental health-related hospitalization were higher following induced abortions than other pregnancies (104.0 vs. 42.0 per 10,000 person-years). Abortion was associated with hospitalization for psychiatric disorders (HR 1.81, 95 % CI 1.72–1.90), substance use disorders (HR 2.57, 95 % CI 2.41–2.75), and suicide attempts (HR 2.16, 95 % CI 1.91–2.43) compared with other pregnancies. The associations were greater for patients who had preexisting mental illness or were aged less than 25 years at the time of the abortion. Abortion was strongly associated with mental health hospitalization within five years but risks waned over time."