Nuclear norm and l1linf norm TFOCS

I have a matrix formulation problem to solve. The matrix variable is a tall thin matrix which should be sparse both row wise and column wise.
I have formulated my problem perfectly in TFOCS. My problem is that
I can regularize my inverse problem solution with either nuclear norm or sparsity norm (l1linf norm). The solution i get with the nuclear norm is better than the solution i obtain with l1linf prox operator. The solutions to l1linf/l1l2 regularizations are identical and slightly worse.
So it would be really nice if i can use both l1linf and nuclear norm regularization at the same time! but it is really hard for me to write my own code! Is there any hope
that the nuclear norm code could use information about sparsity as well soon?
More than that i know another prior that my solution should have positive entries.
With these regularizations i don’t get positive entries. Is there a way also to include this in a prox file? How can i write it in the code?