Let $X$ be a path connected space. If $S$ and $R$ are two equivalence relations on $X$, then one knows that $X/S$ and $X/R$ are path connected. Assume $S \subset R$, i.e $\forall x_{1},x_{2}\in X$, one has $ x_{1}Sx_{2}\implies x_{1}Rx_{2}$. Now suppose $s_{1},s_{2}\in X/S$ ,$s_{1} \ne s_{2}$, are such that there exists a path $\Gamma$ in $X/S$ from $s_{1}$ to $s_{2}$ that maps to a constant path in $X/R$. Does this allow one to extend the relation $S$ to $S'$ in a way that $S\subsetneq S' \subset R$ and $$X/S\cong X/S'?$$
For Example
On $I^{2}$ Let S be given by $(t,0)∼(t,1)$, $(0,t_{1})∼(0,t_{2})$ and $(1,t_{1})∼(1,t_{2})$ ∀t,$t_{1},t_{2}∈I$. Let $R$ identify $∂I^{2}$. Both $X/R$ and $X/S$ are $S^{2}$, and there is a path $Γ$ with said property. There are various extensions $S′$, the maximal one being simply $R$. I am wondering if this is possible even in cases when $X/R$ and $X/S$ need not be homeomorphic, but one can find a maximal $S′$ based on just the existence of $Γ$.