Generalised Sylvester's conjecture for two prime factors

Let $\ell_1$ and $\ell_2$ be distinct primes different from $3$, and let $n$ be one of

$$n=\ell_1\ell_2,\qquad n=\ell_1^2\ell_2,\qquad n=\ell_1^2\ell_2^2.$$

Generalised Sylvester's conjecture. If $n\not\equiv\pm1\pmod9$, then

$$\operatorname{rk}E_{-432n^2}(\mathbb{Q})=\begin{cases}\{0,2\},&\ell_1\ell_2\equiv2\pmod3,\\1,&\ell_1\equiv\ell_2\equiv2\pmod3,\\\{1,3\},&\ell_1\equiv\ell_2\equiv1\pmod3.\end{cases}$$

If $n\equiv\pm1\pmod9$, then

$$\operatorname{rk}E_{-432n^2}(\mathbb{Q})=\begin{cases}\{1,3\},&\ell_1\ell_2\equiv2\pmod3,\\\{0,2\},&\ell_1\equiv\ell_2\equiv2\pmod3,\\\{0,2,4\},&\ell_1\equiv\ell_2\equiv1\pmod3.\end{cases}$$

This is the two-prime specialization of the generalized prediction. The paper proves the relevant Selmer parity and rank bounds, but the displayed exact rank alternatives depend on the stated Tate–Shafarevich parity assumption.