Holography of Information (HoI) and the island formula are two recent approaches to the black hole information loss paradox that yield different Page curves. The difference between the Page curves has been argued to be rooted in the algebra that each approach focuses on. In this paper, we study the candidate algebras for the CGHS model. Making the usual assumptions that are made in HoI literature, we establish HoI for the CGHS model coupled to a massless scalar by constructing the radiative phase space and asymptotic algebras at future null infinity. We prove that the quantum state of the right-moving modes can be recovered from an arbitrarily small neighbourhood of the right future null infinity. We then argue that island reconstruction is incompatible with exact commutativity of the left and right radiative algebras. We discuss the possible distinction between the HoI and island approaches for the CGHS model.