Official Guide - Test 2 - short passage - help me understand

How does the very last sentence connect to the a few sentence preceding it? The “that” here is so confusing to me, is it referring to the modification of the building? It is an important detail because I thought author inclined towards 12th century, but the last sentence mentioned 13th century that threw my logic to the disarray.

Nineteenth-century architect Eugène-Emmanuel Viollet-le-Duc contended that Paris’s Notre-Dame cathedral, built primarily in the late twelfth century, was supported from the very beginning by a system of flying buttresses — a series of exterior arches (flyers) and their supports (buttresses) — which permitted the construction of taller vaulted buildings with slimmer walls and interior supports than had been possible previously.

Other commentators insist, however, that Notre-Dame did not have flying buttresses until the thirteenth or fourteenth century, when they were added to update the building aesthetically and correct its structural flaws. Although post-twelfth-century modifications and renovations complicate efforts to resolve this controversy — all pre-fifteenth-century flyers have been replaced, and the buttresses have been rebuilt and/or resurfaced — it is nevertheless possible to tell that both the nave and the choir, the church’s two major parts, have always had flying buttresses. It is clear, now that nineteenth-century paint and plaster have been removed, that the nave’s lower buttresses date from the twelfth century. Moreover, the choir’s lower flyers have chevron (zigzag) decoration. Chevron decoration, which was characteristic of the second half of the twelfth century and was out of favor by the fourteenth century, is entirely absent from modifications to the building that can be dated with confidence to the thirteenth century.

Yes, it refers to the modifications to the building. But you shouldn´t be confused by this: The author seems still to believe that the flying buttresses, at least for the nave and the choir, date from the 12th century because the buttresses with the chevron decoration cannot have been build in the 13th century.

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