Asymptotic slope conjecture for upper P-positions in Maharaja Nim

Let $k,l\in\mathbb{N}$ with $k<l$, and consider $(k,l)$-Maharaja Nim, obtained from Wythoff Nim by adjoining moves of the form $(k,l)$ and $(l,k)$. An upper P-position is a losing position $(x,y)$ with $x\leq y$. Asymptotic slope conjecture. Every upper P-position $(x,y)$ of $(k,l)$-Maharaja Nim satisfies

$$y=\phi x+O(1),$$

where $\phi$ is the golden ratio. The conjecture predicts that the upper P-positions remain within bounded distance of the line of slope $\phi$, uniformly for every fixed pair $k<l$; the source does not provide evidence of a proof or disproof.