  This paper examines the problem of obtaining a $D(4)$-quadruple by adding a
smaller element to a $D(4)$-triple. We prove some relations between elements of
observed hypothetical $D(4)$-quadruples under which conjecture of the
uniqueness of such an extension holds. Also, it is shown that for any
$D(4)$-triple there are at most two extensions with a smaller element.
