The asymptotic number of copies of $\wedge_s$ in $\{\wedge_k,\vee_l\}$-free families

Let $\wedge_r$ be the poset on elements $a,b_1,\ldots,b_r$ with $a<b_i$ for every $i$, and let $\vee_r$ be its dual. For a family of posets $\mathcal P$ and a poset $Q$, let $La(n,\mathcal P,Q)$ denote the maximum number of copies of $Q$ in a $\mathcal P$-free family ${\mathcal F}\subseteq 2^{[n]}$. The sharp lower-bound conjecture. For any integers $k,l,s$ with $s\le k-1$,

$$La(n,\{\wedge_k,\vee_l\},\wedge_s)=\left(\binom{k-1}{s}\frac{l-1}{k+l-2}+o(1)\right)\binom{n}{\lfloor n/2\rfloor}.$$

The paper proves the same expression as a lower bound by generalizing a construction of Katona and Tarján, and conjectures that this lower bound is sharp.