= Solution
The free-boson worldsheet theory factorizes into independent holomorphic and antiholomorphic sectors. The factors $\partial X^\mu$ contract only in the holomorphic sector and produce one tensor $T^{\mu\kappa\rho}$, while the $\bar\partial X^\nu$ factors independently produce $T^{\nu\lambda\sigma}$. Their product is the closed-string version of left-right factorization, often summarized at tree level as closed-string kinematics being a square of open-string kinematics.
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