= Solution
Write the genus-two surface as
$$
S=T_1\cup_\alpha T_2,
$$
where $\alpha$ is separating and each $T_i$ is a one-holed torus. In each $T_i$, choose two disjoint essential proper arcs $p_i,q_i$ from the boundary to itself that form a cut system, so $T_i\setminus(p_i\cup q_i)$ is a disc. Arrange their four endpoints on each copy of $\alpha$ and glue the boundaries so that
$$
p_1,p_2,q_1,q_2
$$
join cyclically into one simple closed curve $\beta$. After smoothing at the four gluing points, $i(\alpha,\beta)=4$. Cutting along $\alpha$ and these four arcs leaves one disc from each $T_i$, so $S\setminus(\alpha\cup\beta)$ consists of two discs. Hence $(\alpha,\beta)$ is the <filling pair on a closed genus-two surface>.
If their curve-complex distance were at most two, there would be an essential curve $\gamma$ disjoint from both: it would be the intermediate vertex of a length-two path. This contradicts filling. Therefore
$$
\boxed{d_{\mathcal C(S)}([\alpha],[\beta])>2}.
$$
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