When half the plant shows a problem, then the actual problem could indeed be something in the roots. Not necessarily nematodes.
Have you tried fertilizing the plant? Since the side with the problem still produces flowers, but the leaves are looking poor, I would use a product that is reasonably high in nitrogen, moderate in phosphate and potassium, and has trace minerals.
Try a soil applied product that will give you fairly fast results, (Miracle grow, or similar) wait perhaps a week (in reasonably warm weather) then try a foliar applied product. Perhaps the same product, diluted and sprayed on the leaves, especially the under side. Plant leaves have special openings called stoma that can take in nutrients. There are more stoma on the bottom of the leaf in most species.
Results:
If the plant greens up with the soil/root applied ferts then there is nothing wrong with the roots of the plant, but for whatever reason the soil on that side is deficient in nutrients. I would apply a slow release complete fertilizer (including trace minerals) rake it in to the soil and mulch with some compost. You could use 'organic' products if you want.
If the plant does not respond to the soil applied fertilizer, but does green up when you spray the fertilizer on the leaves, there is more likely a root problem or vascular problem fairly low on the plant.
Example might be Verticilium. Nematodes can do this.
Cut off a branch that is poor. Look at the cross section where you cut it off. Is it uniformly cream-white? Or is part of it darker, perhaps tan to brown? Maybe 1/4 to 1/2 of the branch. This is typical of a plant with some sort of vascular problem. Usually a fungal or bacterial disease.
If it is all cream-white, then dig carefully into the soil and look at the roots. Start by looking near the main stem-root junction, but you may also have to work your way out along the roots. Normal roots are relatively smooth, tapering, no major lumps. Look at some pictures of Root Knot Nematode so you know what you are looking for. Several pests can produce roots like this. Nematodes are much more common in sandy soil, but can be found in all soil types.
