The ell^1-grid junta-size conjecture independent of the codomain size

Let $f:\mathbb{Z}_k^n\to\mathbb{Z}_l$ satisfy

$$\sum_{j=1}^n\mathbb{E}_x\left[|f(x)-f(x\mathbin{\oplus}e_j)|'\right]\leq B.$$

Here $|\cdot|'$ is the source's normalized difference notation. Junta-size conjecture. The function $f$ is $\epsilon$-close to an $M$-junta $g:\mathbb{Z}_k^n\to\mathbb{Z}_l$, where

$$M\leq\exp(C_4Bk/\epsilon),$$

for an absolute constant $C_4$. This would remove the dependence on $l$ from the corresponding junta bound; the source gives no resolution.