I don’t have much to add. Like Mark mentioned, the large bounds should raise a red flag.
I would love if you took the time to implement your model in MOSEK Fusion to test the new native exponential cone solver (or perhaps in CVXPY, which should support the exponential cone in MOSEK). We have many examples in the documentation and in the MOSEK Cookbook on exponential cone optimization, and I am convinced that your models will be solved much faster and more reliably if you go that route. If you contact support@mosek.com, you can get a prerelease version of MOSEK 9.0, which is scheduled for beta release soon.