Teleparallel theory as a gauge theory of translations: comments and issues

The Teleparallel Equivalent to General Relativity (TEGR) is often presented as a gauge theory of translations, i.e., that uses \emph{only} the translation group $T_4 = (\setR^4, +)$ as its gauge group. In a previous work we argued against this \emph{translation-only} formalism on the basis of its mathematical shortcomings. We then provided an alternative proposal using a Cartan connection. Recently, a reply by some of the authors defending TEGR as a \emph{translation-only} gauge theory discussed our objections. Here, we first clarify our arguments, and give new proofs of some statements, to answer to these discussions, maintaining our first claim. We then amend one of the argument that originally led us to propose the Cartan connection in this context. This broadens the \emph{a priori} possible choices for a TEGR connection.