The additive height conjecture for two pegs in peg solitaire

Let $A$ and $A'$ be points of a peg-solitaire board, and let $I$ be an initial position. Define $\operatorname{Height}(A,I)$ as the minimum number of legal moves needed to put a peg at $A$ starting with pegs on all points of $I$, with value $\infty$ if this is impossible. Define $\operatorname{Height}(A,A',I)$ analogously as the minimum number of legal moves needed to put pegs at both $A$ and $A'$, again with value $\infty$ if no such succession exists. Additive height conjecture.

$$\operatorname{Height}(A,A',I)\ge\operatorname{Height}(A,I)+\operatorname{Height}(A',I).$$

The claim would give a combined lower bound for the number of moves needed to place pegs at two specified points. The source presents it as a conjecture, and no resolution is supplied in the paper.