A Steiner quadruple system of order v (SQS(v)) is said to be have an almost spanning block design and denoted by 1-AFSQS(v) if it contains a subdesign S(2,4,v−1). In 1992, Hartman and Phelps posed a problem: Show that there exists a 1-AFSQS(v) for each v≡2(mod12). In this paper, we prove that the necessary condition for the existence of a 1-AFSQS(v) is also sufficient with a definite exception v=14 and possible exceptions v∈{86,206,374,398}.