Let $\mathcal{U} = \{ U_i \; |\; i\in I \}$ be an open covering of $X$. Spanier defined $\pi (\mathcal{U}, x)$ to be the subgroup of $\pi_1 (X, x)$ which contains all homotopy classes having representatives of the following type: $ \prod_{j=1}^{n}u_j *v_j * u^{-1}_{j}, $ where $u_j$'s are paths (starting at the base point $x$) and each $v_j$ is a loop inside one of the neighbourhoods $U_i \in \mathcal{U}$.

If an open cover $\mathcal{U}$ is a refinement of an open cover $\mathcal{V}$, then $\pi (\mathcal{U}, x) \subset \pi (\mathcal{V}, x)$.

My question is that:

If $[f][g]\in \pi (\mathcal{U}, x)$ for $[f],[g]\in \pi_1 (X,x)$, then is there any refinement $\mathcal{V}$ of $\mathcal{U}$ so that $[f]\in \pi (\mathcal{V},x)$?