Whether you are doing block coodinate descent or successive convex approximation or the combination of both, too many rounds of iterations with CVX could be unstable due to numerical issues, see the suggestion 6 in how-to-deal-with-an-issue-using-cvx. So you are responsible for the quality of your input data to every subproblem of every iteration.
When your data are too many, if you want to find out which data is bad, the only way might be using MOSEK diretly with MATLAB, not with CVX, so you can locate the bad data by reading the mosek warnings. But the bad news is that now you need to model your convex problem yourself, per https://docs.mosek.com/modeling-cookbook/index.html, without CVX’s help.
I guess you will never try a non-convex solver if you are just writing an academic paper, because the theoretical time complexity is huge, it is not nice looking to put that on your paper. And also, you might not need a global optimal solution in most cases, you need to obtain a low-complexity fine solution with your home-made method, which might be the highlight of your paper.