(CNN) South Dakota Governor Kristi Noem argued in a press conference Tuesday that the explosive outbreak of coronavirus at the Smithfield Foods plant in Sioux Falls could not have been prevented by a stay-at-home order.

As of Tuesday, 438 Smithfield workers in Sioux Falls had tested positive for coronavirus, and the plant, one of the country's largest pork processing facilities, is shut down indefinitely. But Noem said a shelter-in-place order targeted only at the surrounding community would not be coming from her office.

"I've seen some national stories written that a shelter-in-place would have prevented this outbreak at Smithfield. That is absolutely false," Noem said. The governor says that even with a broad order, the plant would have stayed open due to its status as a major food supplier. "This is a critical infrastructure job plant," she said.

Noem is one of a handful of governors who has refused to issue a stay-at-home order, rejecting a request from Sioux Falls Mayor Paul TenHaken.

"That could be a local decision that the mayor and city council could choose to do," Noem said.

Read More