Koszul cohomology of Čech cohomology modules

Let $R$ be a commutative Noetherian ring, let $\mathbf{x}=x_1,\dots ,x_n$ be an $R$-regular sequence, and let $\mathbf{y}=y_1,\dots ,y_m$ be a sequence of elements of $R$. Put $I=(\mathbf{y})$. Let $\mathcal{S}$ be a Serre subcategory of the category of $R$-modules. We consider the double complex obtained from the Koszul co-complex with respect to $\mathbf{x}$ and the Čech complex with respect to $\mathbf{y}$. Using the two spectral sequences associated with this double complex, we prove that $$Ex{t}_R^i(R\mathrm{/}(\mathbf{x}),H_I^j(R))\in \mathcal{S}\text{for all }i,j\in {\mathbb{N}}_0$$ implies $$H_I^j(R\mathrm{/}(\mathbf{x}))\in \mathcal{S}\text{for all }j\in {\mathbb{N}}_0\mathrm{.}$$