Assume $M=3$ and $L=2$ for the multirate system shown below. Furthermore, assume an arbitrary fourier transform $X(e^{j\Omega})$. This one lies between $-\pi$ and $\pi$. The fourier transform after the filter $H_{LP}(z)$, which is an ideal low-pass filter with a cutoff-frequency of $\Omega=\frac{\pi}{3}$, lies between $-\frac{\pi}{3}$ and $\frac{\pi}{3}$, since we sample up by a factor of three. If I then sample down by two, the resulting fourier transform lies between $-\frac{2\pi}{3}$ and $\frac{2\pi}{3}$. Now the spectra of the input and output will clearly look different, therefore also the signal in time. Is it right to say that with this specific multirate system there will be information loss, although no aliasing effects occur?
Note: In case my problem description is confusing, I can provide an example.