Optimality of the constant-deviation density bound for continuous triangulations

Let $\mathcal{T}$ be any locally finite triangulation of $\mathbb{R}^2$ and let $f$ be a continuous piecewise linear $\varepsilon$-approximation of $F(x,y)=xy$ with respect to $\mathcal{T}$. Optimality conjecture. The triangle density satisfies

$$\delta(\mathcal{T}) \geq \frac{\sqrt{3}}{8\varepsilon}.$$

The bound is proved for parallelogram tilings with constant deviation, while the conjecture asserts that allowing arbitrary continuous triangulations and varying deviations cannot improve it. Establishing this would resolve whether non-parallelogram tilings can achieve a lower density.