do most tokyoites know what "yoshiwara" refers to?

It depends really. Some would get what it means, but more would ask back if you live in Shizuoka where there used to be Yoshiwara City (now part of Fuji City). Its postal address “Senzoku” is also the name of a posh residential area in Tokyo (written differently in Chinese characters tho), and wouldn’t ring a bell.
 
Most people know about the area and what goes on there. I got a long lecture by my boss on avoiding the area as well as the two adjacent sections, one of which is where the Yaks house their petty thugs and the other where they own a lot of real estate and keep tabs on the soaps.
 
not unless you imply it is YOSHIWARA since no locales would say I live near yoshiwara as it doesn't exist as a place or name on modern maps in 2017. The proper term is Senzoku

If you say IRIYA, you shouldn't get a cold stare (I think)...
funny how the other areas in Taito-ku, such as Ikenohata or Uenosakuraghi, will give a much more educated impression.
 
Most people know about the area and what goes on there. I got a long lecture by my boss on avoiding the area as well as the two adjacent sections, one of which is where the Yaks house their petty thugs and the other where they own a lot of real estate and keep tabs on the soaps.

He just wants to keep it all to himself, lol
 
Yeah most people know what it is and what goes on there. I wouldnt recommend bringing it up in casual conversation with coworkers.
Probably depends on what company you work in and who your coworkers are. In some cases, they may invite you to a shop that only lets foreigners in when they come together with Japanese :)
 
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Most people know about the area and what goes on there. I got a long lecture by my boss on avoiding the area as well as the two adjacent sections, one of which is where the Yaks house their petty thugs and the other where they own a lot of real estate and keep tabs on the soaps.

Yes, but I don’t think many Japanese know where exactly Yoshiwara is. It’s not a landmark, not even a postal address. Indeed the area between Asakusa and Minami-Senju was sort of no-go because of San’ya, which is bordered by Yoshiwara to the west, where yaks used to manage thousands of laborers living there. Still many Japanese just don’t go to that whole area while gentrification is ongoing. In fact, quite a few day laborers’ hostels in San’ya have been converted to backpackers’ inns. If you say “I live near Yoshiwara.”, another possible response would be “oh you must be a backpacker.”
 
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I got a long lecture by my boss on avoiding the area as well as the two adjacent sections

Yes Boss, I will do as you say. Specifically which houses I should avoid? Can you mark them in this map I happen to have with me, please?