Hereditary algebraicity of sum-closures of ordered binary structures

Let $\mathfrak D$ be a hereditary class of indecomposable ordered binary structures. Suppose that $\mathfrak D$ is hereditary well-quasi-ordered and hereditary algebraic. Sum-closure conjecture. The sum-closure of $\mathfrak D$ is hereditary algebraic. This conjecture would imply that the sum-closure of the class of critical bichains is hereditary algebraic; the latter is known to be well-quasi-ordered and algebraic, but its hereditary algebraicity remains open.