The $l_1$-coherence dominance conjecture for relative entropy of coherence

For a $d$-dimensional quantum state $\rho$, let $C_{l_1}(\rho)$ be its $l_1$-norm coherence and let $C_r(\rho)=S(\operatorname{diag}(\rho))-S(\rho)$ be its relative entropy of coherence, where $S$ is the von Neumann entropy and $\operatorname{diag}(\rho)$ is obtained by deleting the off-diagonal entries of $\rho$.

$l_1$-coherence dominance conjecture. For all states $\rho$,

$$C_{l_1}(\rho)\geq C_r(\rho).$$

The preceding proposition proves the weaker bound $C_r(\rho)\leq\log_2 d\,C_{l_1}(\rho)$, and the conjecture proposes removing the dimension-dependent factor. The source notes that the conjectured inequality is sharp for qubits but gives no general proof.