A universal upper bound for pair discrepancy

Let $D({\bf p})$ denote the total variation discrepancy between the color distributions of a pair selected by the two matching procedures, and define

$$\ell_0:=\sup_{\bf p}D({\bf p}),$$

where the supremum ranges over all probability distributions ${\bf p}$ on a finite or countable set of colors. Universal discrepancy bound. The constant satisfies

$$\ell_0<1.$$

Since total variation distance is at most $1$, this conjecture asks for a nontrivial universal bound on the discrepancy; the paper does not establish it.