Let $S_g$ denote the closed oriented surface of genus $g\geq 2$. Let $x,y$ be two different (upto fixed base point homotopy) but freely homotopic curves, i.e. $y$ is a non-trivial element from a fixed conjugacy class of $x$. If there exist a simple closed loop $z$ (non-trivial) with $i(x,z)\neq 0$ , where $i(\ , \ )$ denotes the geometric intersection number. Then is it true that $i(x*y,z)\neq 0$?
One can prove that if $y=x$ (upto fixed base point homotopy) , then $i(x,z)\neq 0$ implies $i(x^2,z)\neq 0$ , because the hyperbolic axis correspond to the translation $[x],[x^2]\in \pi_1(S_g)$ in $\mathbb H^2$ are same. Also the reason I've chosen $z$ to be a simple closed loop, becasue I think $x*y$ cannot be a simple closed loop here.
I'm unable to find some proof/counter-example. Although my intuition says that it should be true but I would be happy if some one give me a counter-example with a good (general) explanation.