The Candy Nim pile-splitting conjecture

Let $G=[a,b,c]$ be a Candy Nim game with $a<b<c$. For some integer $j>1$, choose positive integers $a_1,\ldots,a_j$ satisfying

$$a=a_1+\cdots+a_j=a_1\oplus\cdots\oplus a_j.$$

Define

$$H=[a_1,a_2,\ldots,a_j,b,c].$$

Candy Nim pile-splitting conjecture. There exist such $j$ and $a_1,\ldots,a_j$ for which

$$V(H)\geq V(G).$$

This conjecture proposes that a 3-pile game can be replaced by a game obtained by splitting its smallest pile while preserving both its total size and nim-sum. The paper explicitly notes that not every such decomposition has this property.