So it's a hook parameter, part of state kept by the tab. Right.

If the hok 'initialize-tab produce that variable local to tab, I cannot see how to send this variable to the hooks 'show-tab or 'kill-tab that uses it, via parameters of hooks.

In fact, it is possible. There is a fact in computation, that using an operation of adding, testing for 0, and having many variables , one can do everything.