Strict ordering of the high-order critical coupling constants

Let $d\geq 3$, and let $(\beta_{L^N})_{N\geq 2}$ denote the critical coupling constants associated with the $N$-th moments of the stochastic heat equation. Strict-ordering conjecture. The high-order critical regimes satisfy

$$\cdots < \beta_{L^{N+1}} < \beta_{L^N} < \cdots < \beta_{L^3} < \beta_{L^2}.$$

This conjectures that the critical coupling constants are strictly decreasing with the moment order. The asymptotics of mixed higher moments in dimensions $d\geq 3$ remain unclear, and the relation between their limits and the spatially averaged solution is not fully understood.