Congruence modules and Wiles defects of determinantal rings of maximal minors

Let $O$ be a discrete valuation ring and $A:=O[X_{m\times n}]\mathrm{/}{I}_m(X)$ the determinantal ring of maximal minors. We consider algebra maps $\lambda \text{\hspace{0.17em}⁣}:A\to O$, which is tantamount to choosing rank-deficient matrices $a\in O^{m\times n}$. Following Iyengar–Khare–Manning, we compute the congruence module and the Wiles defect of $A$ at $\lambda$, expressing them in terms of the $(m-1)$-sized minors of $a$.