Singular Values & the SVD · Superchallenge problem

A real $4\times 4$ matrix $A$ has singular values $6$, $4$, $3$ and $1$. A $4\times 3$ matrix $B$ is formed by deleting one column of $A$. Over all such $A$ and all choices of the deleted column, find the smallest possible value of $\sigma_1(B)\sigma_2(B)\sigma_3(B)$.