Conjectured asymptotic formula for the coherence of random sparse vectors

Fix $N$ and $M$. Let $\boldsymbol{a} \in \mathbb{R}^N$ be a random $K$-sparse vector whose $K$ nonzero entries, on an arbitrary support $S$, are independent and identically distributed Gaussian random variables with distribution $\mathcal{N}(0,\frac{1}{K})$. Let $\rho_c(\boldsymbol{a})$ denote the coherence quantity defined in the paper. Asymptotic coherence conjecture.

$${\bf E}\left[\rho_c\left(\boldsymbol{a}\right)\right] \sim \frac{K}{c_1K+c_2},$$

where $c_1=\frac{1}{c\log L}$ for some constant $c$, and $c_2=1-c_1$. The conjecture proposes a sparsity-dependent behavior that is not captured by the preceding logarithmic upper bound; it is motivated by simulations showing approximately linear growth of $K/{\bar{\rho}_c(\boldsymbol{a})}$. The authors state that they are unable to prove it, while reporting close agreement between the proposed formula and empirical data.