Edgeworth box and identical tastes

If tastes are homothetic and identical, does this imply that the contract curve is the line connecting the lower left and upper right corners of the box?

My understanding: no, it does not. Consumers' indifference curves, though identical, could have the same slope along some line to the side of the line connecting the lower left and upper right corners of the box, e.g. in the case of steep indifference curves.

Is this correct?

Take the situation where $$u_1(x,y)=x+y$$, $$u_2(x,y)=x+y$$. Then the contract curve would be the entire box.