"I usually spend my time in Tut's tomb," said Hiebert about the Egyptian pharaoh Tutankhamun's burial site, "but this is more important."
National Geographic is partnering with Greek restoration experts to document the work.
A 12th-century building sitting on 4th-century remains, the Church of the Holy Sepluchre is the only place where six Christian denominations practice their faith at the same site.
The Edicule was last restored in 1810 following a fire, and is in need of reinforcement after years of exposure to humidity and candle smoke. A hulking iron cage built around the Edicule by British authorities in 1947 for support still stands, but is not enough.
Renovations at this holiest of spots require mutual agreement by the church's various custodians, and that is notoriously hard to secure. The denominations jealously guard different parts of the site and often object to even the slightest of changes.
Last year, Israeli police briefly shut down the building after Israel's Antiquities Authority deemed it unsafe. It prompted the Christian denominations to green light the repairs, which began in June.
Pilgrims line up throughout the day for the chance to crouch in the Edicule's tiny room. They kneel before a white marble encasing, said to cover a surface hewed from the side of the limestone cave where Jesus' body was laid before his resurrection.
Church officials closed the Edicule to pilgrims beginning Wednesday evening, and workers used a pulley to slide open the marble slab, in hopes of reaching the burial surface.
Hiebert said the slab hadn't been removed since the year 1550. Underneath the marble was a layer of debris. By Thursday afternoon, workers had finished removing the debris, revealing something unexpected: another marble slab.
Hiebert said he thinks the second slab, which is grey and features a small etching of a cross, dates to the 12th century. It is cracked down the middle, and underneath it is a whitish layer.
"I don't believe ... that is the original rock," Hiebert said. "We still have more to go."
The main Christian communities that govern the church have allowed the work crew only 60 hours to excavate the inner sanctum, Hiebert said. Experts are working day and night to reach the tomb's core and to analyse it.
"We will close the tomb after we document it," said Antonia Moropoulou, an architect at the National Technical University of Athens, which is supervising the renovation.
The restoration team wants to tightly seal the core of the tomb before injecting parts of the shrine with mortar for reinforcement, so the material doesn't seep inside what is considered to be the holy rock.
One part of the tomb will remain exposed. Experts on Thursday cut a rectangular window in one of the Edicule's marble walls, so pilgrims will be able to glimpse, for the first time, a part of the limestone wall thought to be the tomb of Jesus.
David Grenier, secretary of a group that oversees Roman Catholic church properties in the Holy Land, stood with a few other Franciscan friars, watching the work crew in awe.
"What happened here 2000 years ago completely changed the history of the world," he said. "To be able to dig, let's say, to the rock where the body of Jesus was laid ... it's overwhelming joy." At one point, a National Geographic film crew documented the site as clergy burned incense around them in a daily church rite.
After the film crew cleared out, a pair of clergymen in brown frocks, and an Israeli policeman stationed at the church to help keep the peace, clambered over a pile of work tools, electrical wires and a yellow hard hat on the Edicule floor to lean into the inner chamber and snap cell phone photos of the exposed tomb.
"It's a historic moment, huh?" the policeman said.
