= Solution
The precise conclusion is
$$
\boxed{\mathbb P\!\left(\lim_{n\to\infty}\frac{H_n}{n}=\frac23\right)=1.}
$$
To prove it, observe that the maximum rises exactly when the new height difference is positive:
$$
H_m-H_{m-1}=\mathbf1_{\{D_m>0\}}.
$$
A block landing in a shorter column produces $D_m=0$ and no rise; every other case raises the maximum and leaves $D_m>0$. Starting from $H_0=0$, telescoping gives
$$
\frac{H_n}{n}=\frac1n\sum_{m=1}^n\mathbf1_{\{D_m>0\}}.
$$
The difference chain is irreducible and positive recurrent by part (i). Apply the <countable-state Markov chain ergodic theorem> to the bounded <indicator function> $g(k)=\mathbf1_{\{k>0\}}$. Its stationary mean is $\sum_{k\geq1}\pi_k=1-\pi_0=2/3$, giving the asserted <almost sure convergence>. No <aperiodicity> assumption is needed for this time average. This establishes the <growth rate of two-column sticky deposition>.
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