Extended Knuth conjecture for joint arc-count distributions in random digraph depth-first search

Consider a random multidigraph on $n$ vertices in which the outdegree of each vertex is independently distributed as $\operatorname{Ge}(1-p)$, with $p\in(0,1)$, and endpoints of outgoing arcs chosen independently and uniformly from the $n$ vertices. In a Depth-First Search, let $L,F,B,C,T$ denote the counts of the corresponding arc types used in the paper. Extended Knuth conjecture. For every $n\geqslant1$ and every $p\in(0,1)$,

$$(L,F,B+C,T)\overset{\mathrm{d}}{=}(L,B,F+C,T).$$

This strengthens Knuth's equality in distribution by asserting a joint symmetry involving the other arc counts. It is presented as a conjecture and no resolution is supplied in the source.