The coordinatewise junta approximation conjecture for Lipschitz maps between discrete tori

Let $f=(f_1,\ldots,f_m):\mathbb{Z}_k^n\to\mathbb{Z}_l^m$ be $\alpha$-Lipschitz with respect to the $L^1$ norm. Coordinatewise junta conjecture. For every $\delta,\epsilon>0$, at least $(1-\delta)m$ coordinates $i\in[m]$ have $f_i$ $\epsilon$-close to an $M$-junta $g_i:\mathbb{Z}_k^n\to\mathbb{Z}_l$, where

$$M\leq\alpha\exp\left(\frac{Ck}{\delta\epsilon}\right),$$

and $C$ is an absolute constant. This would replace the exponential dependence on the Lipschitz parameter in the corresponding theorem by linear dependence; the source gives no resolution.