Unitification of weakly Rickart and weakly p.q.-Baer *-rings

S. K. Berberian raised the problem ``Can every weakly Rickart $*$-ring be embedded in a Rickart $*$-ring with preservation of right projections?". Berberian has given a partial solution to this problem. Khairnar and Waphare raised a similar problem for p.q.-Baer $*$-rings and gave a partial solution. In this paper, we give more general partial solutions to both the problems.