I intend for this to be my last post.
From the graph presented, it is evident that the idea can work for methane but under limited conditions, and ultimately, the results are not desirable overall.
It's difficult to transform this graph into one that focuses on iso-octane, or standard grade gasoline. We would have to assume that since the attached electrolysis device requires so much energy, that this new graph would have to show much larger jumps than shown here for methane with a little concentration of hydrogen and oxygen. I can't speculate if this outcome would be true.
If it were true, I would expect that the vehicle must be run very lean (more air relative to stoichiometric ratio). Depending on the leanness of the fuel/air mixture, power loss may occur. So if this device did work, it could only work for such engine conditions, and benefits would only be seen if the driver consistently drove within such engine conditions.
An alternative way to look at the problem is to question if we can use a substance different than water in the electrolysis device to cut back the energy demand to produce hydrogen on board the vehicle. It appears that we can, but the idea is potentially dangerous and also the design of the electrolysis device has durability issues and decomposition issues with the electrocatalysts. Liquid ammonia consumes 95% less energy than water during electrolysis; ammonia requires 1.55 W-h/g relative to water 33 W-h/g to produce 1 gram of H2. If the hurdles associated with the quick electrocatalyst decomposition can be overcome, it makes good sense that we may see this electrolysis technology in the future.