I found the solution to the smell: a Peace Lily. We put it on a plant stand in one corner of the room near the rug. I got it at Green Acres in Roseville. The plant expert advised us to buy it. I didn't believe her but she was convincing. The foul rubber odor is almost completely gone. The plant is among several that act as air purifiers - removes the bad and emits the good. Here's a good source about it: http://www.todayshomeowner.com/best-houseplants-to-improve-indoor-air-quality/
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Rug smell
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