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Mainly motivated by the quaternionic Calabi conjecture raised by Alesker andVerbitsky, in the current note we study the complex $(n-1)$ Monge-Amp\`ereequations on compact Hermitian manifolds. Following the recent strategies ofGuo-Phong-Tong, we give a uniform estimate which depending only on the$p$-entropy of the prescribed volume form and the background datum.

$L^\infty$ estimate for a complex Monge-Amp\`ere type equation from hypercomplex geometry