I doubt the news, even though it would make me very happy. 3850 may be more credible.
While that 150 mAh different looks huge when written that way, but it's really less than a 4% difference to go from 3850 to 4000, which - if we assume a normal user is getting about 6 hours of SOT, would mean about 14 minutes of additional screen on time. Users aren't going to notice that, or care. It'd be maybe another hour of total usage, including idle, which is definitely not noticable. Another way of writing it would be increasing from 3.9a to 4.0a - and when the difference is 0.1/4.0, it's pretty obvious that number is not a big deal and no one would care
Or at least, they shouldn't care about that any more than going from 6.32" to 6.38" on a diagonal display size on a tall heavy shape. It's literally a 1.9% (0.3in^2 or 193mm^2) increase in display area - the device will be 0.03" (0.7mm) wider and 0.06" (1.3 mm) taller.
The Note 8 is 3300 mAh, and in order to make an easily noticeable difference in it's battery life, if going at it from a battery capacity standpoint only, the battery would have to be at least 5200 mAh, if not 6400 mAh. That would be the range to get the device to go from easily making it through one day, to easily getting it through two days, for most users.
What I would like to see is the display changed to 147.66mm x 73.83mm, which is a 6.50" display, which would give an increase in display area of 1.2in^2 or 766mm^2, it would correct the aspect ratio to a true 2:1 and the total dimensions of the display would only be an average of 3.5mm (barely over an 8th of an inch) more in length and width. That, plus flattening the edges, would give users a ton of extra display space, while allowing for (because of the approximately .25" taller and wider the device would have to be to accomodate no "edge" nonsense and the 3.5mm increase), would mean the device has a total area increase of 12.7% - which, when coupled with a .4mm increase in thickness, means that an 18% larger battery can be put in the device without any other changes. Since they were already proposing a 3850 mAh battery, allegedly, I'm going to assume they can do that in a similar chasis to the Note 8, so I'll take my 18% larger on top of that - so 4550 mAh would be the minimum acceptable battery size.