Edit: Sorry about misinterpreting. What my previous argument shows is that there is a counterexample if and only if there is a counterexample for $s=2$.

Unfortunately, there is a counterexample for $s=2$.

$(1-i-j-k)(1+i+j+k)=4=0$ mod $4$.

$(1+i+j+k)i(1-i-j-k)=(1-i-j-k)(i-1+k-j)=j^2 - (1-i-k)^2$

$= -1-1+1+1+2i+2k=2i+2k\neq 0$ mod $4$