The constant-playability relaxation of the playable 1-2-3 conjecture

Let $G=(V,E)$ be a graph without isolated edges. For a function $f:E\rightarrow\mathbb{N}\cup\{0\}$, let $G$ be $f$-unbalanceable when $(G,\emptyset)$ is $(f,A)$-unbalanceable for arbitrarily chosen infinite sets $A\subseteq\mathbb{F}$.

Constant-playability relaxation. Let $G=(V,E)$ be a graph without isolated edges. Then there exists a constant $c\in\mathbb{N}\cup\{0\}$ such that $G$ is $f$-unbalanceable, where

$$f:E\rightarrow\mathbb{N}\cup\{0\},\qquad f(e)=c\quad\text{for every }e\in E.$$

The paper proposes this as a weaker relaxation because no constant bound is known for the online 1-2-3 conjecture problem. The source gives no resolution.