The 3/8 conjecture for 2-tight components in 2-colored 3-graphs

Consider a $2$-edge-coloring of the complete $3$-uniform hypergraph $K_n^3$. Let $M(n,2,3,2,3)$ be the largest number of $3$-sets contained in a monochromatic $2$-tight component, and define

$$\Lambda(2,3,2,3)=\lim_{n\to\infty}\binom{n}{3}^{-1}M(n,2,3,2,3).$$

The 3/8 conjecture.

$$\Lambda(2,3,2,3)=\frac{3}{8}.$$

This is identified in the paper as the smallest unsolved case outside the graph setting. The stated heuristic suggests improved lower bounds, but the exact value remains open.