Suppose I have a compact surface $\Sigma$ in a contact 3-manifold, where the boundary $\partial\Sigma$ is transverse to the contact structure. Am I able to perturb $\Sigma$ rel boundary so that it is convex (in the contact geometry sense)?

If I can't do this rel boundary, what sort of things could I do to perturb $\Sigma$ so that it is convex? For example, I could perturb $\partial\Sigma$ so that it is Legendrian, and then I could pertub $\Sigma$ so that it is convex (assuming $tb(\partial\Sigma)<0$). Would any complications arise if I then perturb this boundary to be transverse after doing this?

I haven't found any explicit mention of these kinds of surfaces, and I am hoping there is just a quick answer to this.