Monotonicity conjecture for geometric-scaled sums

Let ${\boldsymbol X} \equiv \{X_i\}$ be i.i.d. arbitrarily distributed random variables with mean $\lambda$, and let

$$Y_n \equiv \boxplus_{(\frac{1}{n}, \ldots, \frac{1}{n})}{\boldsymbol X}.$$

The preceding central limit theorem states that the p.m.f. of $Y_n$ converges to the geometric distribution of mean $\lambda$ as $n\to\infty$. Entropy monotonicity conjecture. The entropy $H(Y_n)$ increases monotonically with $n\in\{1,2,\ldots\}$ and converges to the geometric entropy ${\cal E}_g(\lambda)$ as $n\to\infty$. This is analogous to the law of small numbers and to the discrete entropic-monotonicity result for ultra-log-concave random variables, but the conjecture is not restricted to ultra-log-concave random variables; the available result only establishes monotonicity when $n$ increases in power-of-$2$ steps.