No, because a reconstruction filter is used to remove the stairsteps. This does not lose any information. I recommend watching the xiph.org videos explaining it:
EDIT: Also, consider that true square/triangle/sawtooth waves are mathematical abstractions that can't exist in reality. If you try to move a real loudspeaker cone in a square wave, you have to reverse direction in exactly zero time. This requires infinite acceleration and therefore infinite force. If you take the Fourier transform of these waveforms you get an infinite series of harmonics.
A real-world "square" wave only contains the lower harmonics within some frequency band. When you limit it to audio frequencies, all square waves above 6.67kHz are identical to sine waves because the only harmonic within that frequency band is the fundamental.
https://wiki.xiph.org/Videos
EDIT: Also, consider that true square/triangle/sawtooth waves are mathematical abstractions that can't exist in reality. If you try to move a real loudspeaker cone in a square wave, you have to reverse direction in exactly zero time. This requires infinite acceleration and therefore infinite force. If you take the Fourier transform of these waveforms you get an infinite series of harmonics.
A real-world "square" wave only contains the lower harmonics within some frequency band. When you limit it to audio frequencies, all square waves above 6.67kHz are identical to sine waves because the only harmonic within that frequency band is the fundamental.