Hello,
First I most say that while TaskBoard can handle large datasets, a kanban board is not the ideal UI for it. It is better suited for a dataset of limited size.
Secondly sorry but no, TaskBoard has not been designed to handle pagination and it wont send individual queries per column.
Currently it has no mechanism to "load on demand", but you can try implementing it in your app. I would probably try listening for the https://bryntum.com/products/taskboard/docs/api/TaskBoard/view/TaskBoard#event-renderTask
event. When it is triggered for the last (or last-ish) task in a column, load more tasks and add them to the store.
If you have very many tasks per swimlane, you might want to consider setting a height on the swimlanes to make them individually scrollable.