Regulated local good-reduction conjecture for A-motives

Let $A$ be the coefficient ring, let $F$ be a finite extension of its fraction field, let $\mathfrak p$ be a finite place of $F$, and let $\underline M_{\mathfrak p}$ be an $A$-motive over $F_{\mathfrak p}$. Let $\ell$ be a maximal ideal of $A$ such that $\mathfrak p$ does not lie above $\ell$. Write $\operatorname{Ext}^{1,\mathrm{reg}}_{\mathcal O_{\mathfrak p}}(\mathbbm{1}_{\mathfrak p},\underline M_{\mathfrak p})$ for regulated extensions in the integral category, and $\operatorname{Ext}^{1,\mathrm{reg}}_{\mathrm{good}}(\mathbbm{1}_{\mathfrak p},\underline M_{\mathfrak p})_\ell$ for regulated extensions having good reduction in the $\ell$-adic realization.

Regulated local good-reduction conjecture.

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

In particular, the module on the right does not depend on $\ell$.

This is the proposed regulated analogue of the local integral-part conjecture for $A$-motives; the source presents it as strongly expected and obtains the corresponding inclusion.