Divisor-transference conjecture for $\Phi_{4p}(2)$

Let $\mathcal{P}_3$ and $\mathcal{S}_3^{(\leq 3)}$ be as used in the paper, and define

$$A(x):=\#\bigl(\mathcal{S}_3^{(\leq 3)}\cap[1,x]\bigr),$$ $$Z(p):=\#\Bigl\{r\mid\Phi_{4p}(2):(r-1)/(4p)\in\mathcal{S}_3^{(\leq 3)}\Bigr\}.$$

Here $\Phi_{4p}(2)$ is the value of the cyclotomic polynomial $\Phi_{4p}$ at $2$. Divisor-transference conjecture. There exists $C>0$ such that for every sufficiently large $p\in\mathcal{P}_3$,

$$Z(p)\leq C\cdot\frac{A(2^{2p}/(4p))}{p}.$$

This is a divisor-level sufficient criterion aimed at controlling admissible prime divisors of the fixed cyclotomic value $\Phi_{4p}(2)$, rather than counting primes by density. It remains open.