A mod 32 class-number congruence for quadratic fields of discriminants $-8p$ and $8p$

Let $p$ be a prime such that $p\equiv 3\pmod 4$. Let $h(-8p)$ and $h(8p)$ denote the class numbers associated with the discriminants $-8p$ and $8p$, respectively, and let $\Psi$ be the function used in the paper. Conjectural congruence. One has

$$h(-8p)\equiv h(8p)\left(\frac{\Psi(2\sqrt{2p})}{3}-\frac{\Psi\left(\frac{1+\sqrt{2p}}{2}\right)}{3}\right)\pmod{32}.$$

The authors report that this stronger congruence holds for every prime $p\equiv3\pmod4$ with $p\leq1000$, whereas the paper establishes a corresponding congruence modulo $16$. Its validity in general remains open.