Approximation by elements in intersection of two Banach subalgebras

Question

Let $A$ be a Banach algebra, and let $A_1,A_2$ be Banach subalgebras of $A$. Suppose that there exists $c>0$ such that whenever $a_i\in A_i$ ($i=1,2$) and $||a_1-a_2||<\varepsilon$, then there exists $b\in A_1\cap A_2$ such that $\max(||b-a_1||,||b-a_2||)<c\varepsilon$. If I pass to the suspensions, in other words I consider $SA=\{f\in C([0,1],A):f(0)=f(1)=0\}$, and also $SA_1$ and $SA_2$, do I still have the same property? It seems that I do if I allow $(c+\delta)\varepsilon$ where $\delta>0$ but can I do it with $c\varepsilon$?