Corrected rank conjecture for twin prime elliptic curves

Let $p\geq7$ and let $(p,p-2)$ be a twin-prime pair. Define the twin-prime elliptic curve by

$$E_p:y^2=x(x-2)(x-p).$$

Write $r(E_p)$ for its Mordell-Weil rank. Corrected twin-prime rank conjecture.

$$r(E_p)=\begin{cases} 0 & \text{if }p\equiv7\pmod8,\\ 1 & \text{if }p\equiv3,5\pmod8,\\ 0\text{ or }2 & \text{if }p\equiv1\pmod8.\end{cases}$$

The first case is established in the cited prior work, and the source proves the analytic-rank lower bound and relevant rank upper bound for the middle cases. The full assertion, especially the exact Mordell-Weil rank in the $p\equiv3\pmod8$ case and the alternatives in the $p\equiv1\pmod8$ case, remains conjectural.