Completeness conjecture for regular Inverse Treblecross positions

Let $R$ be the regular set of board states, let a state have at most one piece, and let $\mathcal{G}$ denote its Sprague–Grundy value. Completeness conjecture. The regular set $R$ contains all board states that possess at most one piece and satisfy $\mathcal{G} \le 1$, with exactly four isolated exceptions:

$$[1 X 1],\quad [1 X 4],\quad [4 X 1],\quad \text{and}\quad [4 X 4].$$

These four states are described as isolated anomalies, and empirical data indicates that all other sufficiently long states in their residue classes have $\mathcal{G}>1$; the completeness of $R$ beyond the stated exceptions remains unproved.