When the DropDownList in the tab is clicked, the tab will be focused, thus the DropDownList will be closing immediately. For this reason, you could set a flag in order to mark whether the DropDownList should close. In the Dojo example linked here the closing of the DropDownList is prevented in the close event handler depending on the closeAllowed flag value. In case the user clicks outside the DropDownList the closeAllowed value is set to false, thus the DropDownList should close correctly.
I would also suggest you take a look at the Kendo Toolbar demos to check if the Toolbar is more suitable in your scenario:
I hope you will find the provided information and example helpful.
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