On conditions for SLLN for martingales with identically distributed increments

For any random variable $X$ with $\mathbf E\big[|X|\log(1+|X|)\big]=\infty$ and $\mathbf{E}X=0$ we construct a sequence $\{X_n:n\ge1\}$ of martingale differences which are identically distributed with $X$ and such that the strong law of large numbers does not hold.