Intersecting families with covering number three II

A family $\mathcal{F}\subset \binom{[n]}{k}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The covering number of a family $\mathcal{F}$ is defined as the minimum size of $T\subset [n]$ such that $T\cap F\neq \emptyset$ for all $F\in \mathcal{F}$. In 1980, the first author proved that for sufficiently large $n$, any intersecting $k$-graph $\mathcal{F}$ with covering number at least three, satisfies $|\mathcal{F}|\leq \binom{n-1}{k-1}-\binom{n-k}{k-1}-\binom{n-k-1}{k-1}+\binom{n-2k}{k-1}+\binom{n-k-2}{k-3}+3$. There was very little progress during more than forty years but recently (cf. \cite{FW25}) with a completely different approach we proved the same result for the full range $n\geq 2k$ and $k\geq 7$. In this short paper we prove the same inequality for all the remaining cases.