If the initial endowment is on the contract curve, then there must always be a competitive equilibrium in which no trade takes place.

Contract Curve

In economics, it's the locus of points representing the final allocation of goods or resources that cannot be improved upon from the perspective of any individual without worsening the condition of another, particularly referenced within the Edgeworth box for pure exchange economies.

Competitive Equilibrium

A market state where supply meets demand, leading to an optimized distribution of goods and services among competing actors.