= Solution
\b[False for directed distance.] Take four <vertices> $a,b,c,d$. Put both directed <edges> between every distinct pair in $\{a,b,c\}$, and add $d\to a$ and $d\to b$. Every <vertex> has <out-degree> two, satisfying $|V|/2$, and there are no <self-loops>. Nevertheless no <directed path> leads from any of $a,b,c$ to $d$, so the claimed finite upper bound on <graph distance> fails.
If “connected” means weakly connected, that portion does follow: a weak component of size $m$ gives each of its vertices <out-degree> at most $m-1$, so each component would have size at least $|V|/2+1$, making two components impossible. It does not imply a <strongly connected directed graph>. Thus either the strong-connectivity interpretation fails outright or the weak interpretation leaves the distance assertion false; the complete claim is false in either case.
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