Best differential characteristic for this PRF
The function $f$ is biased towards the complement of the input $c_{i,j}$, assuming the other two inputs are approximately randomly distributed. 

As all the values $c_{i,j}$ are public, this means that the output of the function $R_i$, and hence the function $F$ is strongly distinguishable from random (being biased towards a known bit pattern). This isn't an answer the specific question you've asked, however it would appear to be a more severe weakness than a high-probability differential would be.