Hybrid semigroup-friable shifted-prime conjecture

Let $\mathcal{S}_3^{(\leq 3)}$ be the bounded-exponent semigroup used in the paper, and let $\operatorname{ord}_r(2)$ denote the multiplicative order of $2$ modulo a prime $r$. Hybrid semigroup-friable shifted-prime conjecture. For all sufficiently large $k\in\mathcal{S}_3^{(\leq 3)}$, no prime $r$ satisfies both

$$\operatorname{ord}_r(2)=4k \qquad\text{and}\qquad \frac{r-1}{4k}\in\mathcal{S}_3^{(\leq 3)}.$$

This is proposed as an analytic target that would close the finiteness problem for $H_{\mathrm{even}}$ by ruling out primitive divisors passing both parts of the 3-Higgs test. It remains open.