Congruences of first syntomic cohomology groups

Let O_K be the ring of integers of a finite extension K of Q_p. Given two reflexive F-gauges on O_K, we show that for large enough n, the mod p^n-reductions of their first syntomic cohomology groups, which might be regarded as a refinement of local Bloch–Kato Selmer groups, are isomorphic if and only if the mod p^{2n}-reductions of their attached Breuil–Kisin modules with G_K-actions and Nygaard filtrations are isomorphic.