The HKOTY conjecture on equality of the M-sum and N-sum

Let $\mathfrak{g}$ be a simple Lie algebra. For dominant integral $\lambda$ and multiplicity data $\mathbf n$, let $M_{\lambda,\mathbf n}^{(k)}$ and $N_{\lambda,\mathbf n}^{(k)}$ be the $M$-sum and $N$-sum, respectively, with the additional restrictions $m_{\alpha,i}=n_{\alpha,i}=0$ whenever $i>t_\alpha k$, where $t_\alpha$ is $1$ for long roots, $2$ for short roots of $B_r,C_r,F_4$, and $3$ for the short root of $G_2$. HKOTY conjecture. For any simple Lie algebra,

$$M_{\lambda,\mathbf n}^{(k)}=N_{\lambda,\mathbf n}^{(k)}.$$

This is a strengthened finite-level form of the conjecture that the restricted $M$-sum equals the unrestricted alternating $N$-sum. The paper proves the stated identity, so the conjecture is resolved.