= Solution
Define the piecewise-linear function
$$
h(s)=g(0)+g'(0)s+
\sum_{i=1}^N\bigl(g'(K_i)-g'(K_{i-1})\bigr)(s-K_i)^+.
$$
On $[K_j,K_{j+1})$, its slope is $g'(K_j)$. Since $g$ is <convex>, $g'$ is nondecreasing and hence $h'(s)\leq g'(s)$ wherever the derivatives exist. As $h(0)=g(0)$, integration gives $h(s)\leq g(s)$ for every $s\geq0$.
Positive no-arbitrage pricing, the forward identity $\mathbb E_{Q^T}[S_T\mid\mathcal F_t]=F_t^T$, and the call-price formula now give
$$
\pi_t\geq B_t^T\bigl(g(0)+g'(0)F_t^T\bigr)
\bigl(g'(K_i)-g'(K_{i-1})\bigr)C_t^{T,K_i}.
$$
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