No screen can change the contrast of a projector. Any screen that has darker blacks will have dimmer whites. I would use a white screen in a bat cave but others may disagree. If the projector is overly bright on such a small screen a gray screen may be a better choice, but any ALR factors will just introduce possibly poor viewing angle, hot spotting and sparkles.
"Smart enough to know better, to old to care" ------ Dedicated Bat Cave Home Theater, JVC RS49U/BenQ 3550 Projectors, 110" 16:9 Jamestown screen with variable power masking for CIW 2.50:1 to 16:9, Denon 4500H with 7.1.4 Atmos with Ohm mains, SVS PB2000, 3 DIY Subs (2 15" (1 ported, 1 sealed and a 12" 4th order bandpass), 1 DIY butt kicker, Custom Built HTPC, 18TB DroboFS NAS