I am familiar with the generic formula to convert OR to Cohen's d. However, I get the very strong impression from a relatively recent paper (https://www.tandfonline.com/doi/abs/10.1080/03610911003650383) that the rate of an outcome in the unexposed group can matter a great deal in this conversion. Depending on the OR value, and the rate, the converted d can vary. Specifically, as the rate increases, the d increases for a given OR.
The authors of that paper did not bother to give their method. How does one take this into account?