That would be easy enough to verify by simply running the browser benchmark with a game active.
EDIT
Also, that benchmark is quite old. Not sure if either MS or Sony have since made performance improvements to their browsers.
EDIT 2
SunSpider 1.0.2 and
Kraken 1.1 both seem to be freely available and accessible. I'll run the test on the XBO and PS4 when I get home, with both an active game and no active game.
I'll update the results as a go along. For now, here's SunSpider on the PS4. Each condition was tested 3 times. Active game was Metro Last Light Redux with a saved game loaded (not at the menu). The game was not "Suspended" for those wondering. Active Video App was Crunchyroll. Crunchyroll will happily continue to play the stream in the background (audio drops out of course). Game was not active or in a suspended state except where noted.
I'll finish testing Kraken and run the same tests on the XBO later (this it taking longer than I thought).
SunSpider 1.0.2
PS4 (No Open Game or App): 918.0 ms =/- 0.7 %
PS4 (No Open Game or App): 916.3 ms =/- 0.5 %
PS4 (No Open Game or App): 931.3 ms =/- 1.1 %
PS4 (Game Actively Running, No Open App): 1111.5 ms =/- 2.6 %
PS4 (Game Actively Running, No Open App): 1076.1 ms =/- 0.4 %
PS4 (Game Actively Running, No Open App): 1081.6 ms =/- 0.5 %
PS4 (Video App Actively Running, No Open Game): 1003.5 ms =/- 1.8 %
PS4 (Video App Actively Running, No Open Game): 1010.2 ms =/- 1.0 %
PS4 (Video App Actively Running, No Open Game): 1020.3 ms =/- 1.0 %
PS4 (Video App Actively Running + Game Suspended): 1058.4 ms =/- 6.9 %
PS4 (Video App Actively Running + Game Suspended): 1019.9 ms =/- 0.5 %
PS4 (Video App Actively Running + Game Suspended): 1030.3 ms =/- 1.7 %
PS4 (Video App In Background + Game Active): 1142.4 ms =/- 0.8 %
PS4 (Video App In Background + Game Active):1144.9 ms =/- 1.3 %
PS4 (Video App In Background + Game Active): 1136.3 ms =/- 1.6 %
Kraken 1.1
PS4 (No Open Game): 20973.6 ms =/- 1.5 %
PS4 (No Open Game): ms =/- %
PS4 (No Open Game): ms =/- %
PS4 (Game Actively Running): ms =/- %
PS4 (Game Actively Running): ms =/- %
PS4 (Game Actively Running): ms =/- %