Achievability of symmetric capacity via equal-amplitude diagonal entries

Let $K\geq 2$ and let $P$ denote the common SNR of a symmetric $K$-user Gaussian multiple access channel. For real numbers $\beta_2,\ldots,\beta_K$ satisfying $|\beta_k|\geq 1$, let $\tilde{L}$ be the matrix defined in the source. The condition of equal-amplitude diagonal entries is

$$|\tilde{L}_{kk}|=|\tilde{L}_{jj}|\quad\text{for all }k,j.$$

Achievability conjecture. There exists a positive number $P^*(K)$ such that, for every $P\geq P^*(K)$, one can choose real numbers $\beta_2,\ldots,\beta_K$ with $|\beta_k|\geq 1$ so that the diagonal entries of $\tilde{L}$ have equal amplitude.

The equal-amplitude condition is the condition used in the paper to achieve the symmetric capacity. The preceding discussion notes that the two-user case requires $P\geq 1.5$, while the general threshold is not determined in the source.