Complex compatibility-degree conjecture for one dichotomic and one $(k+1)$-outcome measurement

Let $k\in\mathbb N$ and $d\geq 2$. The minimum compatibility degree $s_{\mathbb C}(d,2,(2,k+1))$ concerns one dichotomic measurement and one measurement with $k+1$ outcomes on a $d$-dimensional complex quantum system. Complex compatibility-degree conjecture. The minimum compatibility degree is

$$s_{\mathbb C}(d,2,(2,k+1))=\frac{k-1+\sqrt{k+1}}{2k}.$$

This would show that the dimension-two real optimum remains optimal in every dimension and over complex quantum mechanics; the paper reports numerical support for small $k$, but the general claim remains open.