Are consumer systems able to measure nanosecond timings?
If your question is: is there a way with a low cost product to reliably measure a difference of 6 ns?

Then the answer is yes. There are for example time to digital converters which cost less than 1$ which are able to measure time with a resolution of 55 ps. 

They are based on standard CMOS technology I don't think it would be too much trouble to integrate the time measurement into a system.



However I cannot answer the question if that would actually be enough of a safety gain to render the relay attack harmless. The paper you linked suggests, that this might help. 

If the next question you ask: why isn't it done then? Because of cost. If something costs a cent more and the risk is not deemed high enough, it won't be done (that's at least my gathering on automobile industry).