My initial thought was also to use a metal rod (~3/16" dia.) under the treads to pull the sides together. If preferred, a slot (square or preferably rounded. i.e. bull nose router bit) could be milled to the bottom of the tread to accept the rod since the dado will be supporting the tread vertically. Since the sides are 5/4 there is ample thickness to c'bore & plug the holes at each end of the rod. This would add stiffness while hiding the rod.
Thoughts entering one's mind need not exit one's mouth!
As I age my memory fades .... and that's a load off my mind!
"We Live In The Land Of The Free, Only Because Of The Brave"
“The problems we face today are there because the people who work for a living are outnumbered by those who vote for a living."
"Socialism is a philosophy of failure, the creed of ignorance, and the gospel of envy, its inherent virtue is the equal sharing of misery." Winston Churchill