Partition relations on a plain product order type

The goal of this short note is to interest set theorists in the order type ω∗⋅ω1, and to encourage them to work on the question of whether or not the Continuum Hypothesis decides the partition relation τ→(τ,infinite path)2, for τ=ω∗⋅ω1 and for τ=ω1ω+2.