While we do not have a conclusive study, anecdotal evidence along with the studies we do have are all suggesting an infection rate many times the rate of what we have officially recorded. Common sense would arguably tell you that alone. I know it's the general consensus among the health care professionals I know along with the biologist in my family.
Fwiw, the hospital where my wife works is transitioning to a more normal mode of business. The advanced practice nurses that were assigned to manage the Covid ICUs are being sent back to their normal positions. Likewise for the nurses in the ER that were reassigned. The new challenge will likely be the burst of procedures from the backlog created.
What makes you say that Ochsner doesn't have enough tests?