Congruent number conjecture for angles pi/3 and 2pi/3

Let $n$ be a squarefree positive integer. A positive integer is a $\theta$-congruent number if it is the area of a triangle with rational side lengths and angle $\theta$. The residue classes below are taken modulo $24$.

Congruent number conjecture for angles $\pi/3$ and $2\pi/3$. If

$$n\equiv 11,13,17,\text{ or }23\pmod{24},$$

then $n$ is a $\pi/3$-congruent number; and if

$$n\equiv 5,17,19,\text{ or }23\pmod{24},$$

then $n$ is a $2\pi/3$-congruent number.

The conjecture extends the classical congruent-number conjecture to these two angles. The paper gives several results for primes in the asserted residue classes, but the full statements for squarefree integers remain unproved.