How to show the exact time for dates that happen at midnight?

Is there a way to show the times for dates that happen exactly at midnight in a user’s timezone? If a date is set with the time at midnight, the date widget makes it looks like the dates don’t have a time attached.

I’ve tried different date formats, but it looks like if the event happens in the user’s timezone at midnight, the time is removed.

We use the feature so that users can schedule a kind of online gaming event across time zones, but to some users it will look like the events have a specific time and for others, it won’t look like the events have actual times unless they click (which some might not do).

Here’s a screenshot to illustrate what I mean: