Petrosyan's binary-weight conjecture for interval colorings of complete graphs

Let $K_{2n}$ be the complete graph on $2n$ vertices, and let $W(K_{2n})$ be the greatest number of colors in an interval coloring of $K_{2n}$. Write $\left\|n_2\right\|$ for the number of $1$'s in the binary representation of $n$. Petrosyan's binary-weight conjecture.

$$W(K_{2n})=4n-2-\left\lfloor\log_2 n\right\rfloor-\left\|n_2\right\|.$$

This conjecture was proposed after the earlier exact-value conjecture was disproved. The supplied text gives no resolution of this binary-weight formula, so its status remains open.