Keep it to half a day
Nobody concentrates on a call for seven hours. Three and a half hours with a real break is worth more than a full day that people quietly work through.
A shape that works: an hour of the serious business first, the game in the middle, then forty-five minutes of the softer conversation while everyone is still warmed up.
Split into teams before the day
Buy one ticket per team of four to six and send each team its own link and its own call. Mix departments and mix seniority - the whole value is in watching someone you only know from a status update solve something.
Name the teams in advance. Ten minutes lost to "who am I with?" at the start ruins the momentum.
The debrief that is worth having
Bring everyone back and ask two questions: what did somebody know that they did not say, and who ended up leading? Both translate directly to how the team runs a project, and both are much easier to discuss about a padlock than about last quarter.
Compare times if the teams played matched rooms. It gives the whole thing a scoreboard without anyone being appraised.
Picking the game
The Warp Core is three separate parts, so it suits a team that wants to do this more than once - one part per quarter keeps a distributed team in touch. Trapped on a Train is the better single session and the gentler introduction.
Both are £15 per team, per part, playable for 24 hours from the link, with a 60-minute in-game timer.