What is the minimum size of the group containing a subset of elements $S = \{s_1,\dots,s_k\}$ satisfying the following property: There is no $i,t \in \{1,\dots,k\}$ such that $s_i^t = b_1\cdot b_2\cdot \dots \cdot b_t$ where $b_1,\dots,b_t \in S$ and not all $b_j$ are $s_i$.

There is a group of size $O(k^5)$ satisfying the property above. Can we improve this bound to $O(k)$ or $O(k^2)$?

Just an alternative formulation (from Włodzimierz Holsztyński):

Let $\ S\ $ be a subset of an arbitrary finite group $\ G.\ $ An arbitrary element $\ g\in S\ $ is called a geometric mean within $\ S$ $$ \Leftarrow:\Rightarrow\quad\exists_{n\leq |S|}\exists_{x_1\ldots x_n\in S}\ \left(g^n=\prod_{t=1}^n x_t\ \ \&\ \ \{x_1\ \ldots x_n\}\ne \{g\} \right) $$ This question is about the minimal possible $\ g(k)=|G|\ $ such that there is $\ S\subseteq G\ $ which has no geometric mean within $\ S,\ $ and $\ |S|=k.$