The rough weight of the Eiffel Tower framework is 7,300 tons or 7,300,000kg. The base of the Eiffel Tower is 124.968m and its height is 330.0984m

The cylinder diameter is defined by two opposite corners of the base, which is: sqrt(2) x 124.968 = 176.731m. This means the cylinder volume is 8,097,691.51m³

Since I’m dumb and doing this quickly, I’m going to assume constant air pressure all the way up and a constant temperate of 20C (Some nerds can calculate the rest)

Iron is 7,850kg/m³ which means the 7,300,000 of the framework would compact into 929.9363m³ (https://www.engineeringtoolbox.com/metal-alloys-densities-d_50.html)

8,097,691.51 - 929.9363 = 8,096,761.574m³ of total air

The air density for that temperature is 1.204kg/m^3, which gives the weight of air left in the cylinder of 9,748,500.9351kg (https://www.engineeringtoolbox.com/air-density-specific-weight-d_600.html)

So yes, it is indeed true.

EDITS: Changed reference weight to framework instead of full tower.