Unknown behavior in short exact sequence of short exact sequences

Decide whether, in a short exact sequence of short exact sequences in a pointed category, the comparison map w is always a monomorphism and whether the comparison map x is always an epimorphism.

Background

The paper builds the category SES(X) of short exact sequences in a z-exact category X and shows it is itself z-exact. However, limits and colimits in SES(X) are not computed pointwise, leading to subtle phenomena. Exercise 1.10.7 raises a structural question about the morphisms appearing in a short exact sequence of short exact sequences: whether certain induced maps must be mono/epi. The authors mark certain exercises ‘ANK’ to indicate they are open to them.