Completeness conjecture for obtuse rational lattice triangles

Let $T$ be a rational triangle, meaning that its angles are rational multiples of $\pi$. The triangle $T$ is a lattice triangle if its unfolding is a translation surface $(X_T,\omega_T)$ whose $\mathrm{SL}_2(\mathbb{R})$-orbit is closed in its stratum. An obtuse triangle has one angle in $(\pi/2,\pi)$.

Completeness conjecture. The known list of obtuse rational lattice triangles is complete: every obtuse rational lattice triangle is either a member of one of the two infinite families

$$\left(\frac{\pi}{n},\frac{\pi}{n},\frac{(n-2)\pi}{n}\right)$$

and

$$\left(\frac{\pi}{2n},\frac{\pi}{n},\frac{(2n-3)\pi}{2n}\right),$$

or is Hooper's triangle with angles

$$\left(\frac{\pi}{12},\frac{\pi}{3},\frac{7\pi}{12}\right).$$

The acute and right-angled cases are described as fully resolved, whereas the classification of obtuse scalene triangles, those with distinct interior angles, remains the primary open challenge.