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We prove that, for every fixed $M$, these constants are asymptotically contractive: \\[ \\lim_{m\\to\\infty}K_{m,M}=1. \\] More precisely, \\[ 1\\le K_{m,M}\\le A_M^{M/m}m^{(M^2-1)/(2m)}, \\] where $A_M$ depends only on $M$. The argument combines bounded projections onto exact support levels, a random colouring of the active variables, the classical multilinear Bohnenblust--Hille inequality and interpolation with Parseval's identity. 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We prove that, for every fixed $M$, these constants are asymptotically contractive: \\[ \\lim_{m\\to\\infty}K_{m,M}=1. \\] More precisely, \\[ 1\\le K_{m,M}\\le A_M^{M/m}m^{(M^2-1)/(2m)}, \\] where $A_M$ depends only on $M$. The argument combines bounded projections onto exact support levels, a random colouring of the active variables, the classical multilinear Bohnenblust--Hille inequality and interpolation with Parseval's identity. 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