At some point, one of the MOSEK staff members who read this forum should see this thread and be able to disposition this one way or another.
In the meantime, look at section 5.2 of the MOSEK Modeling Cookbook https://docs.mosek.com/MOSEKModelingCookbook-letter.pdf for some ideas.
Here is a somewhat similar, although concave expression, which no one of the forum has really figured out how to do. Log(log(1+exp(x)) objective function