You must add code to manually fire the jQuery event so subsequent dependent drop-downs are properly populated. Here is sample code that works against that site:
HtmlSelect country = Find.ByExpression<HtmlSelect>(
//Scroll the browser so we can visually confirm the drop-down selections
//Manually invoke the jQuery event. You can reuse this line and change the ID to match the applicable control.
HtmlSelect state = Find.ByExpression<HtmlSelect>(
//Short pause to visually confirm the drop-down selections were correctly made
For more information, please read this article
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