Let $A$ and $B$ be two, possibly dependent, random variables, and let $X$ be a random variable independent of $(A,B)$. For simplicity, let's concern ourselves with discrete random variables. Is the following inequality always true?

$$I(A+X : B+X) \geq I(A:B) \label{eq:conj} \tag{*}$$

This is clearly true when $A$ and $B$ are independent, as the RHS is then $0$. At the other extreme, it is also true when $A = B$ since the RHS is $H(A)$ while the LHS is $H(A+X)$. And, $H(A+X) \geq H(A+X~|~X) = H(A~|~X) = H(A)$, where the last equality uses $A\bot X$. On the other hand, I cannot see a proof even when $B = f(A)$ for some deterministic function $f$.

It is not too hard to see if we added $X$ to only one of $A$ or $B$, the mutual information inequality would be flipped. That is, $I(A+X: B) \leq I(A:B)$. Intuitively this makes sense: a random variable plus noise gives less information about another random variable than without the noise. However, when we add (the same) $X$ to both $A$ and $B$ and ask for the mutual information between them, I have no good intuition.

The setting in which \eqref{eq:conj} arose, $X$ is a Bernoulli, and $A$ and $B$ are sums of iid Bernoullis with common elements. More precisely, $X_1, \ldots, X_n$ are iid Bernoullis, and $A = \sum_{i\in S} X_i$ and $B = \sum_{j\in T} X_j$ where $S,T$ are (potentially intersecting) subsets of $\{1,2,\ldots, n\}$. I experimentally verified \eqref{eq:conj} for small $n$.

Any help/pointers would be appreciated.