Ordered continuous bands on the positive real numbers and distribution

Let $\mathbf R_+$ be the space of positive numbers with the ordinary topology and the ordinary order, and let $\ddagger$ be any ordered continuous band on $\mathbf R_+$. We show that there is no cancellative continuous semigroup operation on $\mathbf R_+$ which is distributed by $\ddagger$. Additionally, we show that if $\ddagger$ is not homeomorphically isomorphic to any of three bands called $\it{min}$, $\it{left}$-$\it{zero}$ and $\it{right}$-$\it{zero}$, then there is no cancellative continuous semigroup operation on $\mathbf R_+$ which is distributive over $\ddagger$. Moreover, we show that if $\ddagger$ is homeomorphically isomorphic to any of these three bands, then all cancellative continuous semigroup operations on $\mathbf R_+$ are distributive over $\ddagger$.