The provided example should successfully update the DropDownList data. Also, the refresh method is not required as the read method refreshes the widget as well.
I can suggest checking the network tab to observe if the request is sent to the server and also if the response is containing the new data.
Also, please use the dataBound event of the DropDown to check if it is rebound after the read method is called:
If additional assistance is needed, please provide a fully runnable example reproducing the issue and I will gladly assist.
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