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The quaternionic Calabi conjecture, posed by Alesker and Verbitsky\cite{Alesker-Verbitsky (2010)}, predicts that the quaternionic Monge-Amp\`ereequation can always be solved on any compact HKT manifold. Motivated by thisconjecture, we will introduce a quaternionic version of the Gauduchonconjecture on any compact $SL(n,\mathbb{H})$-manifold, specifically addressingthe existence of quaternionic Gauduchon metrics with prescribed volume form. Wereframe this question as a special case of fully nonlinear elliptic equationsof second order and subsequently establish a uniform estimate for the potentialfunction.

$L^\infty$ estimate for the potential of quaternionic Gauduchon metric with prescribed volume form