Integral criterion conjecture for semilinear wave equations

Consider the Cauchy problem for the semilinear wave equation with nonlinearity

$$F(u)=|u|^{p_S}\mu(|u|),$$

where $p_S$ is the Strauss exponent and $\mu:[0,\infty)\to[0,\infty)$ is continuous, increasing, and satisfies $\mu(0)=0$. For some $\lambda_0>0$, consider the critical integral

$$\int_0^{\lambda_0}\frac{\mu^{p_S}(\lambda)}{\lambda}\,d\lambda.$$

Integral criterion conjecture. Small-data global existence holds if

$$\int_0^{\lambda_0}\frac{\mu^{p_S}(\lambda)}{\lambda}\,d\lambda<\infty,$$

whereas if

$$\int_0^{\lambda_0}\frac{\mu^{p_S}(\lambda)}{\lambda}\,d\lambda=\infty,$$

then the solution blows up at finite time in general. This conjecture proposes a unified integral criterion for the two opposite phenomena. The preceding iteration estimates suggest the criterion, but the general sufficiency and necessity statements are left open in the source.