Is the unimodular row $(x,y,z)$ completable over the ring $({\mathbb Z}/2{\mathbb Z})[x,y,z,y',z']/\langle x^2+yy'+zz'1 \rangle$ ?

Your question is equivalent to the $n=2$ case of what T.Y. Lam calls "Murthy's $(a,b,c)$ problem" in his book "Serre's problem on projective modules. (This is statement 5.7 on p. 323 in the 2006 edition). Lam indicates that this is wide open. 

