> So instead of this, tried to show history completions only after
> typing a special key.  TAB can't be reused because in some minibuffers
> such as ‘M-!’ (shell-command), TAB completes the command/file names.
>
> There is ‘C-tab’ bound to file-cache-minibuffer-complete.
> When file-cache is not used, then ‘C-tab’ could be rebound
> to a command that completes on the minibuffer history.

Actually two keys are needed: one to complete on the history,
and another to complete on the list of default values.
So maybe like the ‘up’ key navigates through the history,
‘C-x up’ could complete on the history, and like ‘down’ key
navigates through the future history, ‘C-x down’ could
complete on it: