On integral $\mathrm{Ext^2}$ between certain Weyl modules of $\mathrm{GLn}$

Consider partitions of the form $\lambda=(a,1^b)$ and $\mu=(a+1,b-1)$,\\ where $a+1>b-1$. In this paper, we determine the extension groups $\mathrm{Ext}_A^2(K_{\lambda}F,K_{\mu}F)$, where $F$ is a free $\mathbb{Z}-$module of finite rank $n$, $K_{\lambda}F$ and $K_{\mu}F$ are the Weyl modules of the general linear group $GL_n(\mathbb{Z})$ corresponding to $\lambda$ and $\mu$, respectively, $A=S_\mathbb{Z}(n,r)$ is the integral Schur algebra and $r=a+b$.