= Solution
The conditional <Jensen inequality> states that for an integrable $X$ and convex $\phi$, whenever the terms are integrable,
$$
\phi(\mathbb E[X\mid\mathcal G])
\leq\mathbb E[\phi(X)\mid\mathcal G]
\quad\text{almost surely}.
$$
For differentiable $\phi$, the supporting-line inequality $\phi(x)\geq\phi(m)+\phi'(m)(x-m)$ with $m=\mathbb E[X\mid\mathcal G]$ gives the result after conditional expectation. Approximation by supporting affine functions proves the general convex case.
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