In fact a weaker version of Aleksandrov's theorem is true for subharmonic functions. Since $\Delta u$ is a Radon measure, the following result follows from Proposition 4.4 in [1].
Theorem. If $u:\Omega\to\mathbb{R}$ is subharmonic and locally integrable, then for almost all $x\in\Omega$ there exist a vector $Du(x)$ and a matrix $D^2u(x)$ such that for all $1\leq p<n/(n-2)$ ($1\leq p<\infty$, if $n=2$) $$ \left(\frac{1}{|B(x,r)|}\int_{B(x,r)}|u(y)-u(x)-u(x)(y-x)-\frac{1}{2}(y-x)^TD^2u(x)(y-x)|^p\right)^{1/p}\\=o(r^2). $$
I believe, the matrix $D^2u(x)$ is symmetric, but it is not explicitly stated in [AG].
As a corollary, the authors prove Imomkulov's theorem (see the answer of user111), but they do it independently (21 years later!) since they do not quote the work of Imomkulov.
If you want to compare it with the statement of Aleksandrov's theorem, see Second order differentiability of convex functions.
[AG] G. Alberti, S. Bianchini, C. G. Stefano, On the $L^p$-differentiability of certain classes of functions. Rev. Mat. Iberoam. 30 (2014), no. 1, 349–367.