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So the rules state for jink is as follows

"Models in this unit have a 5+ invulnerable save against ranged attacks. In your Movement phase, if this unit Remains Stationary, it loses this invulnerable save until the start of your next Movement phase"

If a player with jink goes second then on the opponents turn one they should have a 5++.
This because they have not remained stationary in their movement phase because they have not had one.
Do I get this right or is there something else in play that I am missing?

You've got that correct. You have a benefit (5++) and retain it until you meet a condition that takes it away.

Since you haven't yet had a movement phase, you have not yet had the opportunity to meet the condition (In your Movement phase, this unit Remains Stationary) in order to lose your invulnerable save.

Thank you.

I thought so but better to get a second opinion.