Exceptional case of the non semi-simple Real Abelian Main Conjecture

In the papers: "The Chevalley--Herbrand formula and the real abelian Main Conjecture (New criterion using capitulation of the class group),J. Number Theory 248 (2023)" and "On the real abelian main conjecture in the non semi-simple case, arXiv (2023)", we consider real cyclic fields $K$ and primes $\ell \equiv 1 \pmod {2p^N}$ totally inert in $K$, implying implicitly, $K \cap \mathbb{Q}(\mu_{p^\infty}^{})^+ = \mathbb{Q}$. In this complementary work, we examine the non linearly disjoint case.