Forbidden mass interval conjecture for semilinear wave equations in de Sitter space

Let $n$ denote the spatial dimension, let $m$ be the mass parameter, and let $\alpha\in(0,\infty)$ satisfy condition ($\mathcal L$). The Cauchy problem is considered with small initial data for the semilinear wave equation in de Sitter space.

Forbidden mass interval conjecture. The interval

$$(\sqrt{n^2-1}/2,n/2)$$

is a forbidden mass interval for small-data global solvability of the Cauchy problem for every $\alpha\in(0,\infty)$ satisfying condition ($\mathcal L$).

The conjecture concerns the obstruction to global solvability for small data in the indicated mass range. The cited prior work and the present paper neither prove nor disprove it; the massless case $m=0$, corresponding to global existence for the semilinear wave equation in de Sitter space-time, is still open.