I would like to customize Tooltips appearing over tasks, but I can't add all the information I want.
From the context of the Tooltip Template I'm able to get the Title, Percentage and Start\End of the tasks, but I can't get extra information from my model.

Please expose the model in the context of the Tooltip Template.

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ADMIN EDIT

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For the time being, there are several options to display additional information from your model in the Tooltip. See details and examples here: Access Model Fields in the Gantt Timeline Tooltip.