Sharpness conjecture for the radio-k-number of even cycles

Let $n$ be even and $k$ be odd with

$$\frac{n}{2} \leqslant k < n-3.$$

Set

$$h=\frac{n-k-1}{2},\qquad p=\gcd(n,h).$$

Write $\langle h\rangle$ for the subgroup generated by $h$ in the relevant cyclic group. Sharpness conjecture. If $n/2\in\langle h\rangle$, then

$$\operatorname{rn}_k(C_n)=\operatorname{LB}(n,k)+p-1.$$

The paper notes that the values of the radio $k$-number are known for most parameters in the stated range, but cases with different parities remain open. The conjecture asserts that the upper bound from the preceding theorem is sharp whenever $n/2\in\langle h\rangle$.