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

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