Diffusivity exponent conjecture for self-repelling Brownian polymers

Let $E(t)={\mathbf E}(X(t)^2)$, and suppose that for some $-1<\alpha<1$ the infrared bounds

$$C_1:=\varlimsup_{p\to0}|p|^{-\alpha}\hat b(p)<\infty,\qquad C_2:=\varliminf_{p\to0}|p|^{-\alpha}\hat b(p)>0$$

hold. Diffusivity exponent conjecture. Under these conditions, the true asymptotic order is

$$E(t)\asymp t^{\frac{4}{3+\alpha}}.$$

The conjecture refines the upper and lower bounds proved in the paper and agrees, for $\alpha=0$, with the previously conjectured order $E(t)\asymp t^{4/3}$; the supplied status evidence indicates that it has been resolved.