Irreducibility of a morphism and indecomposability of a module in a minimal MCM approximation

Question

I've been self-studying Cohen-Macaulay rings using the book by Leuschke-Weigand, and had a question which didn't seem to be answered in the book, but where the results seem to be used implicitly. My terminology is taken from their book, and I'll provide page references.

Throughout, let $(R,\mathfrak{m},k)$ be a local ring. Suppose that

\begin{align*} 0 \to Y \to X \xrightarrow{q} \mathfrak{m} \to 0 \end{align*}

is an MCM approximation of the maximal ideal $\mathfrak{m}$, and moreover assume that this approximation is minimal (see pages 251-252). Consider the map $\hat{q}: X \to R$ obtained from $q$ by composing with the inclusion $\mathfrak{m} \to R$.

1. Is the module $Y$ necessarily indecomposable?
2. Is the map $\hat{q}$ an irreducible morphism? (See page 300.)

I think these facts are needed to prove the converse of 12.27 Proposition in the book, which doesn't seem to be proved (maybe it's obvious!).