 Solution

False. Let the directed edges be $r\to u$, $u\to r$, and $r\to v$. Start depth-first search at $r$ and explore its neighbor $u$ before $v$. ThenThe directed path $u\to r\to v$ exists and $d[u]<d[v]$, but $u$ and $v$ are siblings in the discovery tree. An existing path may leave the current recursive branch via a gray ancestor and only later enter a different branch. The usual white-path condition would need all vertices of a suitable path to be white when $u$ is discovered; that condition is absent here.