Consider the following real- valued random vectors
(1) $X:=(X_1,...,X_k)'$ of dimension $k\times 1$
(2) $W_1:=(X_1,...,X_k, Z)$ of dimension $(k+1)\times 1$
Let $\beta \in \mathbb{R}^k$ and $\gamma \in \mathbb{R}^{k+1}$.
Consider $E(X|X'\beta)$ and $E(X|W'\gamma)$. Is it true that $E(X|X'\beta)=X$ while $E(X|W'\gamma)\neq X$?
The first assertion, that $E(X\mid X'\beta)=X$, is not true. The conditional expectation should be a function of $X'\beta$, and none of the $X_1,\ldots,X_k$ is expressible as such a function (except in trivial cases of $\beta$). For example if the $X_1,\ldots,X_k$ are iid, and if we take $\beta=(1,\ldots,1)'$, then $X'\beta=\sum X_i$ and $E(X_j\mid X'\beta)=E(X_j\mid\sum X_i)=\frac1k\sum X_i$, which is not equal to any of the $X_i$.