For a product of curves X = C1 × … ×Cn over a p-adic field k, in 2 we proposed a conjecture that the kernel of the Albanese map for X is p-divisible when the base field is absolutely unramified and proved this under some assumptions. In this note, we report that when the Jacobian varieties of such curves C1, …, Cn all have good ordinary reduction, the Albanese kernel for the product X = C1 × … × Cn is still p-divisible even if the base field is not unramified but its ramification is small enough.
Toshiro Hiranouchi (2023) studied this question.