Inequality in Frankl's conjecture

Question

For the minimal counter-example to union closed sets conjecture, we have the lower bound $\mid$$\mathcal{A}$$\mid$ $\geq$ $4q-1$ ($\mathcal{A}$ denotes the minimal counter-example family, $q$ denotes the number of elements in $\cup$$\mathcal{A}$). Is there any better lower bound? Is there any research/development happening towards this direction?

Answer 1

There is this recent paper that gives a better bound of: $$\mid\mathcal{A}\mid \geq 4q+1$$