Scholl's integral-part conjecture for motivic extensions

Let $F_{\mathfrak p}$ be a local field, let $M_{\mathfrak p}=h^{i-1}(X)(n)$ for a smooth projective variety $X$ over $F_{\mathfrak p}$ and integers $n,i\geq1$, and suppose that $X$ has a regular model $\mathcal X$ over $\mathcal O_{\mathfrak p}$. Define $\operatorname{Ext}^1_{\mathcal O_{\mathfrak p}}(\mathbb{1},M_{\mathfrak p})$ as the inverse image, under the expected $K$-theoretic isomorphism, of the image of $(K_{2n-i}(\mathcal X)\otimes_{\mathbb Z}\mathbb Q)^{(n)}$ in $(K_{2n-i}(X)\otimes_{\mathbb Z}\mathbb Q)^{(n)}$. Let $\operatorname{Ext}^1_{\mathrm{good}}(\mathbb{1},M_{\mathfrak p})_\ell$ denote the subspace of extensions having good reduction.

Integral-part conjecture. For any prime $\ell$ not lying below $\mathfrak p$,

$$\operatorname{Ext}^1_{\mathcal O_{\mathfrak p}}(\mathbb{1},M_{\mathfrak p})=\operatorname{Ext}^1_{\mathrm{good}}(\mathbb{1},M_{\mathfrak p})_\ell.$$

This conjecture identifies the $K$-theoretic and realization-theoretic definitions of the local integral part of motivic cohomology and supersedes the preceding independence assertion.