Can Carrier Offset cause Image Problems The main reason why we use the Direct conversion receiver with zero IF is that we don't have to deal with the problem of images. For achieving zero IF we need to ensure that the receiver Local Oscillator (LO) is working at the same frequency as the carrier. But I was wondering if there is an offset between the carrier and the receiver oscillator (due to reference mismatch or Doppler shift etc.) do we still have the image problem?
For instance, if the offset is equal to the channel spacing $f_{off} = f_{ch}$, then both the $f_c \pm f_{ch}$ fall at $f_{off}$ after  downconversion mixer. Basically, the offset acting as an IF.
If so, do we still need image reject filters for direct conversion architecture and what then is the benefit of direct conversion architecture.

What I mean is explained in the figure below. We have a carrier frequency $f_c$ but due to some offset we send our data at $f_c+\Delta f_c$. Some other user is sending his data at $f_c - \Delta f_c$ (the image sideband). But after demodulation both the desired and undesired signal components fall back at $\Delta f$
Having images of a signal is very different from having a frequency offset in the signal. Having an image means the signal spectrum is replicated at two or more places.Having a frequency offset in a signal means the spectrum is just shifted in frequency.

First let us look at it from the point of view of the innocent receiver who does not know that such a catastrophe has occurred that the offset itself is equal to the channel frequency. Simply speaking the systems would break down with IQ phase reversed.

If however we have realistic frequency offsets like that of Doppler shifts which for typical cellular frequencies in current wireless is of the order of few hundred Hz then this is usually measured at the receiver and the clock is adjusted. This offset is usually measured by correlating time domain samples after ADC in baseband, since a linear shift in frequency domain equals phase shift in time domain.