Conjecture on the growth and limiting behavior of hypercube domination parameters

Let $\chi_n$ denote the parameter defined in the paper for the $n$-dimensional hypercube. Growth and limit conjecture. The sequence $\chi_n$ satisfies both of the following properties:

1. It is strictly increasing on each interval

$$2^m \leq n \leq 2^{m+1}-2.$$

2. Its limit superior is

$$\limsup\limits_{n \to \infty} \chi_n = \frac{1}{2}.$$

The conjecture predicts a structured monotonicity pattern for the minimal domination parameters and identifies $1/2$ as their asymptotic upper-limit value. The supplied context does not indicate that either assertion has been resolved.