Non-periodicity conjecture for rows of the three-pile Sharing Nim nim-value table

Let the rows of Table 1 consist of the nim-values $\mathcal{G}(0,a,n)$ as $n$ varies, with the corresponding fixed row parameter $a$.

Non-periodicity conjecture. Rows of Table 1 are not ultimately periodic.

The bottom row and the main diagonal are shown to be not ultimately periodic, while the analogous assertion for all rows is presented as a conjecture motivated by this phenomenon and by the dependence of a row on lower rows.