There are infinitely many monotone games over $L_5$

A notion of combinatorial game over a partially ordered set of atomic outcomes was recently introduced by Selinger. These games are appropriate for describing the value of positions in Hex and other monotone set coloring games. It is already known that there are infinitely many distinct monotone game values when the poset of atoms is not linearly ordered, and that there are only finitely many such values when the poset of atoms is linearly ordered with 4 or fewer elements. In this short paper, we settle the remaining case: when the atom poset has 5 or more elements, there are infinitely many distinct monotone values.