"They think you're digging under their houses," Stahl told Doron Spielman while in one of the tunnels.
"It does at points go underneath homes. Deep underneath the ground. Which is why we have these metal supports around us," he replied.
"But it's under Arab homes," she said.
"It's under Jewish homes, Arab homes, and a road," Spielman said. "What concerns these residents, Lesley, is not the tunnel. It's where the tunnel's going. It's what the tunnel means that concerns them."
"Well, where's the tunnel going?" she asked.
"The tunnel one day will open into the Western Wall Plaza. Then you will have undergone an experience that shows the Jewish Temple was important 600 years before Muhammad," Spielman said.
To understand why this is so explosive, you have to understand the geography. Silwan lies in the shadow of some of the holiest sites in the world: Judaism's Western Wall, and Islam's Dome of the Rock and al-Aqsa Mosque.
"There's a feeling of encroachment. The Arab's feel it," Stahl remarked.
"There's no other place in the world that Jews want to live in more than here. The Arabs have Mecca, they have Medina and they may also be interested in Jerusalem. But for the Jews, this is our only home," Spielman said.
That feeling of Jewish encroachment has been heightened by the mayor of Jerusalem, Nir Barkat, who is doing all he can to make sure East Jerusalem remains under Israeli sovereignty.
"We have to maintain a united Jerusalem," Barkat told a crowd at a function we attended.
The mayor brought Stahl to a hilltop over Silwan to show her his latest project, called "King's Garden."
"This is the most important area in the world," he said. "And in the valley right there below us is where King's Garden was."
He wants to create a Bible-themed garden and turn it into a tourist park adjacent to the City of David. But as with the dig, the local Arabs see this as another attempt to gobble up their side of Jerusalem. Building the mayor's park requires demolishing 22 Arab homes in Silwan.
"If you began to demolish these houses, it would be explosive, wouldn't it?" Stahl asked.
"That's why you have to be very smart and sensitive dealing with an area so important in the city of Jerusalem," the mayor said.