Kleitman's conjecture for central families

Chv\'{a}tal conjectured that a star is amongst the largest intersecting subfamiles of a finite subset-closed family of sets. Kleitman later strengthened Chv\'{a}tal's conjecture, suggesting that maximal intersecting subfamilies of $2^{[n]}$ when naturally embedded into $\mathbb{R}^{2^{[n]}}$ take on a particular form. We provide a construction which succeeds in expressing certain families as required by Kleitman's conjecture. We then provide a partial characterization of these families, showing central and certain near-central families in particular to be amongst them.