Semigroup log-mass conjecture

Let $\mathcal{P}_3$ be the set of odd primes used in the paper whose shifted-prime closure has the stated 3-Higgs property, and let $\mathcal{S}_3^{(\leq 3)}$ be the associated bounded-exponent semigroup. Semigroup log-mass conjecture. There exists an absolute $\delta>0$ such that for all sufficiently large odd primes $p\in\mathcal{P}_3$,

$$\sum_{\substack{r\text{ prime},\ r\leq 2^{2p}+1\r\equiv 1\pmod{4p}\\(r-1)/(4p)\in\mathcal{S}_3^{(\leq 3)}}}\log r\leq (2\log 2-\delta)p.$$

This conjecture is intended to create a deficit against the primitive logarithmic mass of $\Phi_{4p}(2)$ and thereby prove finiteness of $H_{\mathrm{even}}$. It is presented as open and strictly weaker than the hybrid conjecture.