On the divergence of the $(C,1)$ means of double Walsh-Fourier series

In 1992, Móricz, Schipp and Wade proved the a.e. convergence of the double $(C,1)$ means of the Walsh-Fourier series $\sigma _{n}f\to f$ ($\min (n_{1}, n_{2})\to \infty , n=(n_{1},n_{2})\in {\mathbb{N}} ^{2}$) for functions in $L\text{log}^{+} L(I^{2})$ ($I^{2}$ is the unit square). This paper aims to demonstrate the sharpness of this result. Namely, we prove that for all measurable function $\delta :[0,+\infty ) \to [0,+\infty ) , \, \lim _{t\to \infty }\delta (t)=0$ we have a function $f$ such as $f\in L\text{log}^{+} L\delta (L)$ and $\sigma _{n}f$ does not converge to $f$ a.e. (in the Pringsheim sense).