Potentially strongly semistable, not strongly semistable vector bundles on an arithmetic surface

Motivated by the p -adic Simpson correspondence, we provide examples of smooth projective curves C of positive genus g over a complete discrete valuation ring of characteristic 0 and residue field F̄ₚ F ¯ p of positive characteristic p≥ 2g p ≥ 2 g and of stable vector bundles E E over C which have potentially strongly semistable reduction, but not strongly semistable reduction over C .