Tightness of the chromatic-number upper bound for H-intersecting graph families

Determine whether there exists a graph H other than a single edge for which the upper bound 2^{\binom{n}{2}-(\chi(H)-1)} is attained by a largest H-intersecting family of graphs on n vertices.

Background

The paper proves that any H-intersecting family of graphs on a common n-vertex set has cardinality at most 2^{\binom{n}{2}-(\chi(H)-1)}. It also observes the lower bound obtained by taking all graphs containing a fixed copy of H, whose size is 2^{\binom{n}{2}-|E(H)|}.

The question asks whether the chromatic-number-based upper bound can be exact for any non-edge graph H. The subsequent discussion only compares the two bounds and explains that they differ unless H is an edge; it does not resolve whether some other construction can attain the upper bound.