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$$S_i-S_{n-i}^*S_n=D_{S_n}X_iD_{S_n}~~{\\rm{and}}~~S_{n-i}-S_{i}^*S_n=D_{S_n}X_{n-i}D_{S_n},$$ where $X_i,X_{n-i}\\in \\mathcal B(\\mathcal D_{S_n}), ~{\\rm{for ~all~}} i=1,\\ldots,(n-1),$ with numerical radius not greater than $1,$ for a $\\Gamma_n$-contraction $(S_1,\\ldots, S_n).$ We construct a conditional 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