Ok, here is a counterexample with the index set $\{1,2\}$ (you can easily extend it to whole $\mathbb N$ if you wish).
Let $(Y_1,Y_2)$ be independent Bernoulli($1/2$) and set $$ (X_1,X_2) = \begin{cases} (Y_1,Y_2), & Y_1 + Y_2>0,\\ (0,1), & Y_1 = Y_2 = 0. \end{cases} $$
In this case $$ \mathrm P(X_1 = 1\mid X_2 = 1) = \frac{P(X_1 = 1, X_2 = 1)}{P(X_2 = 1)} = \frac{1/4}{3/4} = \frac13<\frac12. $$