On the Auslander--Reiten Theory for Extended Hearts of Proper Connective DG Algebras

We prove that, for a proper connective dg algebra $A$ with cohomology concentrated in degrees between $1-d$ and $0$, the extended heart $\mathcal{D}^{fd}(A)^{(-d,0]}\subseteq \mathcal{D}^{fd}(A)$ is an extriangulated category with almost-split conflations. We also prove a version of the 1st Brauer--Thrall Conjecture in this context.