Parini's sharp Cheeger inequality conjecture for triangles

Let $\triangle$ be a triangle, let $\lambda_1(\triangle)$ be its first Dirichlet eigenvalue, and let

$$h(\triangle)=\frac{|\partial\triangle|+\sqrt{4\pi|\triangle|}}{2|\triangle|}$$

be its Cheeger constant. Parini's triangle conjecture. For any triangle $\triangle$,

$$\frac{\lambda_1(\triangle)}{h(\triangle)^2}\geq\frac{4\pi^2}{\left(3+\sqrt{\pi\sqrt{3}}\right)^2}\approx1.3885,$$

where equality holds if and only if $\triangle$ is equilateral. This is the triangular specialization of the conjectured regular-polygon minimization, and the source gives no evidence that it has been resolved.