Is nomitative "that" omittable?

[teleostomi]

We hand the scepter of power to the candidate (that) we trust will do right.

I saw this sentence with "that" or "who" omitted like above. Is "that" or "who" omittable?

Without "we trust" inserted, "that" is not omittable, right? What about this case?

[Pete]

We hand the scepter of power to the candidate (that) we trust will do right.

I saw this sentence with "that" or "who" omitted like above. Is "that" or "who" omittable?

Without "we trust" inserted, "that" is not omittable, right? What about this case?
Yes, the sentence sounds correct with "that" omitted. I think that most people would include "that" (or "who"), however.

Definitely if the sentence did not include "we trust", it would need the relative pronoun "that" or "who" to introduce the relative clause.