Answer
No, the rods would not be necessary.
Work Step by Step
Each link in a hanging chain is supported by its neighbors, so the tension is exactly tangent to the chain's shape. When inverted into an arch, this catenary shape ensures that gravitational compression all along the arch will be similarly tangent to the arch, holding it together. There are no force components that would tend to bow the arch outward so the rods are not needed.
This is discussed on page 234, and Figure 12.14 illustrates this very well.
