Welcome to the new and improved clunky junky Google Chrome. I used to love Chrome. But... anything that forces caching is my enemy. Used to have to only deal with shady ISP's like your cable company trying to cache everything to save bandwidth. Now, it's even more difficult with Google being stingy with bandwidth. What do they have to lose from people using bandwidth anyway? Maybe it's because of Google Fibre. If that's the case, then they're just as stingy as the cable companies they talk shit about.
Anyway, I found a way that works sometimes. If you append a different URL for the style sheet every time, it helps. Not 100% because Google still recognizes the URL and pulls a cache of the actual styles. Cheap. Junky. Clunky. Chrome. I suppose maybe if you did this for every page of the site.. appended a random variable, that it might work better.
<link rel="stylesheet" href="/templates/tjohnsj3/css/style.css<?php echo '?ver='.$cache;?>" type="text/css" />