The asymptotic bicommutative-system bound conjecture

For integers $k\geq 2$, let $n=4k+7$ and $m=2k+3$. Let $\mathrm{Spec}(k,2)$ be the minimum cardinality of a $(2k+3)$-bicommutative system in $\mathcal{P}_{n}(m)$. Asymptotic bicommutative-system bound conjecture. There exists a real number $\lambda$ satisfying

$$0<\lambda<\frac{1}{2}$$

and

$$\lim_{k\rightarrow\infty}\frac{\mathrm{Spec}(k,2)}{\binom{4k+6}{2k+2}}<\lambda.$$

The preceding construction gives the upper bound $0.5625$ for this normalized limit; the conjecture asks whether the asymptotic ratio can be bounded by some constant strictly below $1/2$.