Bounded-row conjecture for three-pile Sharing Nim nim-sequences

For integers $a\geq 1$, consider the nim-sequence

$$(\mathcal{G}(0,a,n))_{n\geq a}.$$

Bounded-row conjecture. Given $a\geq 1$, the nim-sequence $(\mathcal{G}(0,a,n))_{n\geq a}$ is bounded.

The conjecture is based on computation and is presented as an open question about the distribution of nim-values in the rows of the table; no proof or resolution is given.