Pro-representability of $K^M$-cohomology in weight 3 generalizing a result of Bloch

We generalize a result, on the pro-representability of Milnor $K$-cohomology groups at the identity, that’s due to Bloch. In particular, we prove, for $X$ a smooth, proper, and geometrically connected variety defined over an algebraic field extension $k\mathrm{/}\mathbb{Q}$, that the functor $${\mathcal{T}}_X^{i,3}(A)=\ker (H^i(X_A,{\mathcal{K}}_{3,X_A}^M)\to H^i(X,{\mathcal{K}}_{3,X}^M)),$$ defined on Artin local $k$-algebras $(A,{\mathfrak{m}}_A)$ with $A\mathrm{/}{\mathfrak{m}}_A\cong k$, is pro-representable provided that certain Hodge numbers of $X$ vanish.