I've been on that road in October for a hunting trip. I pulled a 29 foot FW with a 4x4 truck. No problem at all going in. Almost didn't make it out due to a minor rain storm. If the road is dry, you should have no problem at all. As soon as it gets a little wet, it becomes thick deep gumbo style mud. The mud will build up on your tires and fill the tread up so you have ZERO traction. If you step out, you'll get 4-5 inches taller from the mud sticking to the bottom of your boots.
If you're going in April, Northern Arizona can still be hit by a late winter storm. You can have rain, snow and all kinds of mud issues. Even if it hasn't stormed for a while, any snow melt can cause havoc.
So if you decide to go, keep an eagle eye on the weather. If there is even a hint of moisture, get out while you can. May and June might be a better time to go since we're usually between our winter and summer rainy periods then.