Keep quiet (3,5)
'keep quiet' is the definition.
(to pipe down is to be quiet)
This is all the clue.
(Other definitions for pipe down that I've seen before include "Sh!" , "Say no more!" , "Shut up!" , "Be quiet!" , "make less noise" .)
(Other definitions for lie doggo that I've seen before include "Stay motionless or hidden - as canine might?" , "Hide and keep quiet" , "Stay motionless or hidden in canine manner" , "Wait concealed" , "Remain hidden in supine and canine position (3, 5)" .)