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In 2010 de Jong proposed a $p$-adic version of Gieseker's conjecture: if $X$is a smooth, simply connected projective variety, then any isocrystal on $X$ isconstant. This was proven by Esnault and Shiho under some additionalassumptions. We show that the conjecture holds in the case of a non-propervariety with trivial tame fundamental group and for rank 1 log extendableisocrystals.

A note on rank 1 log extendable isocrystals on simply connected open varieties