True or False: In a tie-breaking penalty kick shootout, designated kickers may change order after all eligible kickers have kicked once.

In the context of a penalty kick shootout tied at the end of regular play, it is indeed true that designated kickers may change their order after all eligible kickers have kicked once. This rule allows teams the flexibility to adjust their strategy based on the performance of their players or the situation at hand.

After each team has taken their initial series of kicks, each coach might choose to rearrange the order to maximize their chances of scoring or to protect their lead, if they have one. This adaptability can be critical in high-pressure scenarios where the game is on the line.

The other options do not support the standard rules regarding penalty shootouts. There is no requirement for mutual agreement to change the order post-initial kick, nor do the rules stipulate any restrictions on order changes during the shootout itself. Thus, the statement affirming that kickers may change the order after everyone has participated once is accurate.