Critical half-space problem in the complementary exponent range in dimension seven

Determine whether the critical half-space problem with boundary value $c=c_p$ admits bounded positive non-one-dimensional solutions, or is rigid with solution $w_0(x_N)$, for the complementary range of exponents in dimension $N=7$ where the sign condition used in the construction is not established as positive.

Background

For $N=7$, the paper constructs bounded positive non-one-dimensional solutions whenever $\kappa(p)>0$, and proves that this holds for all sufficiently large exponents, including every $p\ge2$ and some interval below $2$. It also proves that $\kappa(p)<0$ for exponents sufficiently close to $1$.

The authors do not resolve the half-space problem in the complementary exponent range where their positive-coefficient construction does not apply. The unresolved issue is whether rigidity or nonuniqueness holds there.