Timon of Athens
This will be a judgement call on your part. Overall, the trail is pretty firm. It is also smoother than a gravel roads because the chat is much smaller than they tend to use on roads these days. (Seriously Kansas, is the budget so bad that you have to throw medium sized rocks onto your roads?) I have talked to many people who have made the trek with a road bike and had no problems.
However, often after rains you will find soft spots where a road bike would sink in a bit. I’m not talking quicksand here but you will definately get a tug on your wheel. If you are alert this is no problem, but a surprise can lead to a comedy routine of overcompensation followed by tragedy. YMMV.