Metric Kaplansky conjecture for bounded metric groups

Let $K$ be a finite field, let $\varepsilon>0$ and $N\in\mathbb{N}$. For a metric group $(G,\|\cdot\|)$ with $\|\cdot\|\leq 1$, let $K[G]$ be its group ring, let $l(a)$ denote the relevant length of an element, let $\Delta(G)$ denote the augmentation ideal, and let $\|\cdot\|_S$ denote the seminorm used in the source. Metric Kaplansky conjecture. There exists $\delta(\varepsilon,N)>0$ such that, for all $a,b\in K[G]$ with $l(a),l(b)\leq N$ and $ab-1\in\Delta(G)$,

$$\|ab-1\|_S<\delta(\varepsilon,N)\quad\Longrightarrow\quad\|ba-1\|_S<\varepsilon.$$

This is a quantitative form of Kaplansky's direct finiteness conjecture; the source says it is known for the permutation metric-group class, but gives no general estimate for $\delta(\varepsilon,N)$.