The optimal fifth-power bound for commuting pairs in finite real sets

Let $\mathcal{A}\subset\mathbb{R}$ be a non-empty finite set, and let $T(\mathcal{A})$ denote the quantity counting the relevant commuting pairs of $2\times 2$ matrices associated with $\mathcal{A}$. Optimal fifth-power bound. For every $\varepsilon>0$, one has

$$T(\mathcal{A})\ll_{\varepsilon}|\mathcal{A}|^{5+\varepsilon}.$$

The preceding theorem proves a bound with exponent $5+c$ for some absolute $c>0$ in the more general setting of finitely supported probability measures. The conjecture asks for the endpoint exponent $5$, up to an arbitrary $\varepsilon$, and is motivated by the paper's other bounds for commuting matrices and by connections with growth in groups and sum-product estimates.