Now to throw a spanner in the works.
Ambient light or not please consider a few parameters before any choice is made.
Reflective light levels are probably the most important thing to consider, the typical aim for a dark room is 15~20ftl(new bulb), we easily end up at around 12 with an aged bulb. The reason I state this is that a grey screen absorbs valuable light.
These slopes are from a AE900, what you are looking at is loss of top and total dynamic range when using various shades of grey by calculated percentage loss.The second graph is why we use a grey screen, to iliminate and absorb the lower end light and create the ilusion of deeper blacks.
Nothing wrong with a deep grey or any grey, just remember your total reflective light has to cut above the ambient level that is directly striking the screen.
The homework is;
find out what the fl output of your projector is for the screen size you are aiming at with a gain of 1. A N9 try gain of 0.8 then N8 at 0.6(Projectorcentral has a good calculator)
Dispite a bright PJ it is easy to get caught by taking things a tad too far. Always allow head room for aging bulbs, think 25%. Also there can be tricks to get lumens high in projectors, when you calibrate them they can lose some of those rated levels.
Push the parameters of the calculators to the extremes, find out where you cant go. Find your sweet spots, work from there.
Changing the balance of colour or shade of neutral around the screen changes your perception of black.
White can still be used if you choose carefully the surrounding shade.
How and what type of lighting is used can make a difference.