Stability of di-extension conditions through SES(X)

For a z-exact category X, determine whether: (i) if X is homologically self-dual, SES(X) is also homologically self-dual; (ii) if dinversion in X preserves normal maps, it also does so in SES(X); (iii) if X is di-exact, SES(X) is di-exact. The authors suspect the answer is negative at least for item (iii).

Background

The paper shows SES(X) is z-exact and homologically rich, but certain properties (HSD, DPN, DEx) may not be preserved when passing from X to SES(X). The exercise asks for a precise analysis of this inheritance.