= Solution
An <Einstein-Hilbert action> contains two derivatives, so its cubic graviton vertex produces only the $k\times k$ part of the product of the two $T$ tensors. The string amplitude also contains cross terms of order $\alpha'k^4$ and a term of order $\alpha'^2k^6$. These cannot arise from pure Einstein gravity. They require higher-curvature corrections, schematically
$$
S_{\rm eff}=\frac1{16\pi G_N}\int d^{26}x\sqrt{-g}
\left[R+\alpha'c_2R_{\mu\nu\rho\sigma}^2
+\alpha'^2c_3R^3+O(\alpha'^3\nabla^2R^3,\alpha'^3R^4)\right],
$$
with index contractions and coefficients fixed by string amplitudes up to <field redefinitions>. The full bosonic-string effective action also contains the dilaton and Kalb-Ramond field. Thus Einstein gravity is only its leading low-energy approximation.
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