On the maximum size of 2-weakly compatible split systems

We consider a Turán-type problem arising in phylogenetics: determining the maximum size of a 2-weakly compatible split system. This compatibility condition arises in the reconstruction of phylogenetic networks from quartet weights. It was previously shown that a 2-weakly compatible split system has size at most $$3\left(\genfrac{}{}{0px}{}{n}{4}\right)+\left(\genfrac{}{}{0px}{}{n}{2}\right)\mathrm{.}$$ We prove that the maximum size is $O(n^{5\mathrm{/}2})$.