Pair-distribution conjecture for the two-nut caching game

Assume $k=j=2$, so the hider caches two nuts and the searcher aims to find both. A hiding strategy is represented by pairs $(y_1,y_2)$ of depths, with the hole placements randomized as described in the source. Pair-distribution conjecture. For every $n$ and $h$, there exists an optimal hiding strategy that is a probability distribution over such pairs, with the two holes selected uniformly from the ${n+1\choose2}$ choices and the two possible depth assignments symmetrized; if the same hole is selected twice, one nut is placed at depth $1$. This is presented as a simplifying structural conjecture; the source says it does not seem difficult but gives no proof.