Strict hierarchy of critical couplings for the fermionic multi-particle model

Let $d$ and $q$ be in the regime considered above, with $q<2$, and let $\alpha_c^{(N)}$ denote the critical coupling for the fermionic $N$-particle problem. Define

$$\alpha_c^{(\infty)}=\inf_N\alpha_c^{(N)}.$$

Strict-hierarchy conjecture. If $q<2$, then

$$\alpha_c^{(1)}>\alpha_c^{(2)}>\alpha_c^{(3)}>\cdots>\alpha_c^{(\infty)}>0.$$

The preceding theorem establishes positivity of $\alpha_c^{(\infty)}$ and the strict inequality $\alpha_c^{(2)}<\alpha_c^{(1)}$, but the strict decrease for all subsequent particle numbers, including $\alpha_c^{(\infty)}<\alpha_c^{(2)}$, remains unproved.