IMO, these types of tests have legally mandated cutoff values. For example, poppy seeds have minute amounts of opiates. They are used in baking. Will eating a lemon tart with poppy seeds make you fail a drug test? This is the kind of question you are asking.
The answer to poppy seeds is: they can be detected, but the interpretation of the result is defined legally. As is how the test is performed. A cutoff lower limit is mandated. Any value below that is considered negative.
EtG can remain in hair for a long period -
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3927158/
EtG can be detected reliably in hair up to 12 weeks, again subject to several "if" statements.
The answers depend on EXACTLY what the legal parameters are - not necessarily pure science. In these tests a 'false positive' is defined. There can be be false negative results in many kinds of assays like this due to the limits of detection:
https://www.ncbi.nlm.nih.gov/pubmed/20975547
There are a lot of tonsorial potions like mouthwash and after shave (Old Spice kinds of products) that have ethyl alcohol in them:
Look for the word "ethanol":
https://householdproducts.nlm.nih.gov/cgi-bin/household/brands?tbl=brands&id=16003196
So unless you can be really specific about what we are talking about any answer we give will be a complete guess. Simon's explanation is good. Ethanol will diffuse to the point where humans can no long detect it in a room.
How is the test performed? Hair? Blood sample?
What is the cutoff for the defined lower limit of detection ?
There MUST be somebody who knows this, like a lab technician at wherever you get tested, and can answer your question. We can only guess.