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In this note, we study a special case of the $4$-pt non-vacuum classicalblock associated with the $\mathcal{W}_3$ algebra. We formulate the monodromyproblem for the block and derive monodromy equations within the heavy-lightapproximation. Fixing the remaining functional arbitrariness using parametersof the $4$-pt vacuum $\mathcal{W}_3$ block, we compute the $4$-pt non-vacuum$\mathcal{W}_3$ block function.

Example of the $4$-pt non-vacuum $\mathcal{W}_3$ classical block