I read the article you linked to with interest, but my personal experience causes me to doubt if all the arguments presented there are correct. For much of my adult life I ate a form of Mediterranean diet, but heavily weighted towards fruit, which I love and found extremely convenient. I considered it a healthy form of fast food. Recently I discovered that I have a problem with raised blood sugar and I am convinced that Bluetit is correct in saying that we diabetics need to give up most forms of fruit, which I have done (not without regrets). Giving up fruit and other high carb options has somewhat lowered my blood glucose, but has done nothing to lower my triglycerides, which were already vanishingly low during all the years I was eating lots of fruit, and now that I am eating almost no fruit are slightly higher, though still well below the targets set. So I am not convinced that fruit and carbs in general are responsible for your wife's high triglycerides.
I see in the article that it is claimed that when we eat fruit it cannot be used for energy but is stored as fat. I find this hard to believe, both because I became very thin while I was still eating lots of fruit, and because fruit is very generally eaten by runners and other sportspeople precisely to give them energy. If it could not do this, I think they would have noticed. My conclusion, for what it is worth, is that your wife could certainly lower her blood glucose levels by giving up fruit, and also by limiting other high carb foods. Whether that will lower her triglycerides is more doubtful, but it is worth experimenting.