Let $(W^1,W^2)$ be a two-dimensional Wiener process with correlation $\rho$. Let $\mathcal{F}^1_t$ be the filtration generated by $W^1$ up to $t$.
I would intuitively think that for $h$ measurable and integrable we have $$ E[h(W^2_T) \lvert \mathcal{F}^1_T] = E[h(W^2_T) \lvert W^1_T] , $$ since $W^1_T$ contains all the information in the path. I am though unable to prove it. Is it true and how do I go about showing it?