I have followed Lumosity for quite some time and the other company Posit Science. In fact, I have the complete Posit Science set of exercises.
Neither have been developed for or studied as a treatment for mTBI or PCS. They are part of the Brain Training industry that is focused on exploiting the population of people who are afraid of developing Alzheimer's Disease. They have value at improving some cognitive and visual skills but there is no evidence they help with mTBI or PCS.
As one who has relearned cognitive skills lost to PCS, I have decades of experience. Sudoku will be as effective at helping with the cognitive skills as it focuses on multiple step process thinking.
BTW. I score very high on both the Lumosity and Posit Science tasks but still have very little immediate and short term auditory (verbal) and visual memory functions.
I think the value of both systems is in developing skills that have been under-developed. These skills can then be used to work-around and accommodate the persistent PCS symptoms. Learning better memory skills and visual skills can go a long way to over-coming the symptoms of PCS.
By better memory skills, I mean better ways of looking at things that need to be memorized. For example, I textualize and/or verbalize visual memories so I can retain them. When driving and at a stop sign, I need to look to the left then repeat, "No traffic on the left" while I look to the right. My visual memory will not remember if there was any traffic on the left so this trick allows me to remember by a different means.
The Posit Science program is problematic in that it does not accommodate persons who struggle with visual clutter. I have tried to suggest they improve their user interface but to no avail.
Both are missing a scheduling system to remind the person to do the exercises each day. It is like buying a membership at a health club. As long as your dues are paid, they don't care if you show up or not.
If you have the funds to pay for these systems without sacrificing other needs, there is no reason to not try them. Just don't expect miracles.
fyi, The Alzheimer's reason is not entirely valid. Such exercises may postpone the onset of AD but not by much. In fact, some researchers think they just delay the slow onset to a later quick onset. An example would be: Rather than spending five years slowly developing the symptoms of AD, the subject has two years with less symptoms then a quick decline over next the three years. The end date is likely unchanged. The current AD meds do the same. I lost my father and my wife's father to AD/dementia. Those with PCS are up to 18 times more likley to develop a AD like dementia.
My best to you all.