A remark on Ext groups for motives with maximal unipotent radicals

Let $\mathbf{T}$ be a neutral tannakian category over a field of characteristic 0. Let $M$ be an object of $\mathbf{T}$ with a filtration $0=F_0M\subsetneq F_1M\subsetneq \cdots\subsetneq F_kM=M$, such that each successive quotient $F_iM/F_{i-1}M$ is semisimple. Assume that the unipotent radical of the tannakian fundamental group of $M$ is as large as it is permitted under the constraints imposed by the filtration $(F_\bullet M)$. In this note, we first describe the $Ext^1$ groups in the tannakian subcategory of $\mathbf{T}$ generated by $M$. We then give two applications for motives, one involving 1-motives and another involving mixed Tate motives, leading to some implications of Grothendieck's period conjecture.