Support bound for higher nim-values in three-pile Sharing Nim

Consider a three-pile Sharing Nim position $(0,a,b)$ with $a\leq b/2$, and let $\mathcal{G}(0,a,b)$ be its nim-value.

Support-bound conjecture. If $\mathcal{G}(0,a,b)=g\geq 2$, then

$$a\leq 2g-1.$$

This is an empirical estimate based on computations: nim-values $g\geq 2$ appear to be distributed among the bottom $2g$ rows or fewer. The source gives no proof or resolution and notes that the bound may be an overestimate.