= Solution
Let $p_K=C(T,K+1)-2C(T,K)+C(T,K-1)>0$ and $D_K=C(T+1,K)-C(T,K)$. Given $S_T=K$, the next value is one of $K-1,K,K+1$. The <martingale> property makes its upward and downward conditional probabilities equal; their sum is the conditional <variance> of the increment, $\sigma^2(T,K)=2D_K/p_K$. Thus
$$
\boxed{\mathbb P(S_{T+1}=H\mid S_T=K)=\begin{cases}D_K/p_K,&H=K-1\text{ or }K+1,\\1-2D_K/p_K,&H=K,\\0,&\text{otherwise}.\end{cases}}
$$
The <discrete Dupire equation> therefore recovers all positive-probability one-step conditional transitions from the <European call option> price surface. No <Markov property> is required: these are conditional probabilities given the current value, not necessarily given the whole past.
\b[Recovery is impossible at a zero-probability state.] For example, $S_t\equiv S_0$ satisfies every stated assumption, but at $T\geq1$ a different integer $K$ can satisfy $|K-S_0|\leq T$ while $p_K=0$. Its conditional transitions can be specified arbitrarily without altering $C$. Thus the literal all-states claim needs the positive-probability qualification.
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