Is it possible to show that every 1-design $D$ with $\lambda=4,k=4$ on $v$ points contain some 1-design $Q$ with $\lambda=1,k=3$ on $v$ points such that every block of $Q$ is some block of $D$ that one of its elements is removed?

Is it possible to show that every 1-design $D$ with $\lambda=4,k=4$ on $v$ points (for $v$ that is a multiple of $3$) contain some 1-design $Q$ with $\lambda=1,k=3$ on $v$ points such that every block of $Q$ is some block of $D$ that one of its elements is removed?

Is it possible to show that every 1-design $D$ with $\lambda=4,k=4$ on $v$ points contain some 1-design $Q$ with $\lambda=1,k=4$ on $v$ points such that every block of $Q$ is some block of $D$ that one of its elements is removed?

Is it possible to show that every 1-design $D$ with $\lambda=4,k=4$ on $v$ points contain some 1-design $Q$ with $\lambda=1,k=3$ on $v$ points such that every block of $Q$ is some block of $D$ that one of its elements is removed?