You should be able to get it working with the following steps:
- In your markup definition file, change the CSS file name on line 111 (assuming no other changes) to include "?v=2" after it, like this:
<Style name="ContentSliderBase" href="resources/content-slider-base.css?v=2"></Style>
- After this, log in to your site as an administrator and click the "Edit Slide" link on your Banner Slider. This forces the markup definition cache to clear, which will update it with the "?v=2" in the markup output, which in turn will cause your browser to request the CSS file again instead of pulling it from cache.
Hope this helps!