Is the Erdős–Faber–Lovász conjecture open still?

Question

Is the Erdős–Faber–Lovász Conjecture open still? According to Wikipedia it is unsolved still, but I think this is not hard to solve this conjecture.

Conjecture: If $n$ complete graphs, each having exactly $n$ vertices, have the property that every pair of complete graphs has at most one shared vertex, then the union of the graphs can be colored with $n$ colors.

My attempt: Consider the, $n-1$ complete graphs $K_n$ which all has a common vertex, say $u$. then this graph can be colored with $n$ colors. by choosing $n-1$ vertices (one from each graph $K_n\backslash \{u\}$) by different color and construct a graph with this vertices and a new vertex, say $v$. we can assign to $v$ color of $u$. Why this is not correct or maybe I don't understand the problem ?

Answer 1

I don't think you understand the problem. It's not just $n-1$ complete graphs with one vertex common to all of them. Each pair of complete graphs can have at most one shared vertex, but different pairs can have different shared vertices. So for $n=4$ you could have the graph pictured: the $K_4$s are circled and the five central vertices are each shared by two of the complete graphs. As $n$ increases, the number of possible configurations increases rapidly.

Answer 2

On January 2021 The problem was settled for sufficiently large values of $n$ https://arxiv.org/abs/2101.04698