On locally compact shift-continuous topologies on semigroups ${\mathcal{C}}_{+}(a,b)$ and ${\mathcal{C}}_{-}(a,b)$ with adjoined zero

Let ${\mathcal{C}}_{+}(p,q{)}^0$ and ${\mathcal{C}}_{-}(p,q{)}^0$ be the semigroups ${\mathcal{C}}_{+}(a,b)$ and ${\mathcal{C}}_{-}(a,b)$ with the adjoined zero. We show that the semigroups ${\mathcal{C}}_{+}(p,q{)}^0$ and ${\mathcal{C}}_{-}(p,q{)}^0$ admit continuum many different Hausdorff locally compact shift-continuous topologies up to topological isomorphism.