Naor's lattice quotient embedding conjecture

Let $c_1(X)$ denote the infimal bi-Lipschitz distortion of a metric space $X$ into $L_1$. For a lattice $\Lambda\subset\mathbb{R}^r$, equip $\mathbb{R}^r/\Lambda$ with the quotient $\ell_1$ metric

$$d(x+\Lambda,y+\Lambda):=\inf_{k\in\Lambda}\|x-y+k\|_1.$$

Naor's lattice quotient conjecture. There exists a constant $c>0$ such that for every $r\in\mathbb{N}$ and every lattice $\Lambda\subset\mathbb{R}^r$,

$$c_1(\ell_1^r/\Lambda)\lesssim r(\log r)^c.$$

This is presented as an $L_1$ analogue of a result of Haviv and Regev. The source supplies no resolution status beyond calling it a conjecture.