Uniform deficit conjecture for 1-cross intersecting set pair systems

Let $m_n(*, *, 1)$ denote the maximum size of a $(1,1,1)$-? set pair system on an $n$-element ground set under the notation of the paper. Uniform deficit conjecture. There exists a positive constant $\varepsilon$ such that

$$m_n(*, *, 1)\le(1-\varepsilon){2n\choose n}$$

for every $n\ge 2$. This predicts a uniform positive gap below the central binomial coefficient for these systems; the source presents it as a belief, and no resolution is supplied.