I have $n$ **IID** Bernoulli random variables denoted by $X_1,X_2,\ldots X_n$ with parameter $p$.

I am interested in knowing if the following inequality involving mutual information holds :

   $\boxed{\max_{p} I(X_1+X_2+...+X_n;Y)\leq \max_{p} I(2X_1+X_3+...+X_n;Y) \leq ...\leq \max_{p} I(nX_1;Y)}$

Note that $p$ is variable and can be different across different mutual information terms in the inequality.

Here, $Y$ is a binary random variable taking values in $\{0,1\}$. Also, $Y=0$ with probability $\frac{1}{x+5}$ when the input, ie, the first argument in the mutual information $I(X;Y)$ takes a value $x$.

**Note:** In fact, I conjecture that the inequality even holds for any decreasing transition probability in the place of $\frac{1}{x+5}$.

Can someone help me provide some insights or ways of proving these? In fact, my numerical simulations tend to agree with the inequality. Any help is appreciated.