Conjecture on cubic-root values of the quadratic-residue product $S_p(x)$

Let $S_p(x)$ be the polynomial defined in the source, let $\omega$ be a primitive third root of unity, let $p>3$ be a prime with $p\equiv3\pmod4$, and let $h(-p)$ be the class number of the imaginary quadratic field of discriminant $-p$. Write $\left(\frac{\cdot}{p}\right)$ for the Legendre symbol, and let $(x_p,y_p)$ be the least positive integer solution of

$$3x^2+4\left(\frac p3\right)=py^2.$$

Then

The cubic-root product conjecture.

$$S_p(\omega^{\pm1})=(-1)^{(h(-p)+1)/2}\left(\frac p3\right)\frac{x_p\sqrt3\mp y_p\sqrt p}{2}\times\begin{cases}i^{\pm1}&\text{if }p\equiv7\pmod{12},\\(-1)^{|\{1\le k<\frac p3:(\frac kp)=1\}|}(i\omega)^{\pm1}&\text{if }p\equiv11\pmod{12},\end{cases}$$

and

$$S_p(\bar\omega)=(-1)^{(h(-p)-1)/2}\left(\frac p3\right)\frac{x_p\sqrt3+y_p\sqrt p}{2}\times\begin{cases}i&\text{if }p\equiv7\pmod{12},\\(-1)^{|\{1\le k<\frac p3:(\frac kp)=1\}|}i\bar\omega&\text{if }p\equiv11\pmod{12}.\end{cases}$$

The source presents this as an open conjecture about exact values of $S_p(x)$ at roots of unity; examples are given for $p=79$ and $p=227$.