The Sprague–Grundy conjecture for square rectairs

A rectair is the rectair family used in CRIM, and $\mathcal G$ denotes its Sprague–Grundy value. Let $r$ and $k$ be integers with $0\leq k<r$. Square-rectair conjecture.

$$\mathcal G(R_{r,r}^k)= \begin{cases} 0& r\text{ is even or }k<r-1,\\ 1& r\in\{3,5\},\ r\text{ is odd, and }k=r-1,\\ 2&\text{otherwise.} \end{cases}$$

The conjecture extends the known values for several rectair families, but the source provides no proof or resolution.