Let $\mathbb{H}=\{q=t+xi+yj+zk:t,x,y,z\in\mathbb{R}\}$ be the quaternions and $f:\mathbb{H}\to\mathbb{H}$ be a function satisfying $\overline{\partial_c}f=0,$ where $$\overline{\partial_c}=\frac{1}{2}\left(\frac{\partial}{\partial t}+\frac{\operatorname{Im}q}{r}\frac{\partial}{\partial r}\right), \ \operatorname{Im}q=xi+yj+zk, \ r^2=x^2+y^2+z^2.$$ Then we have $\overline{\partial_l}\Delta f=0$, where $$\overline{\partial_l}=\frac{1}{2}\left(\frac{\partial}{\partial t}+i\frac{\partial}{\partial x}+j\frac{\partial}{\partial y}+k\frac{\partial}{\partial t}\right), \ \Delta=\frac{\partial^2}{\partial t^2}+\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Is this statement true? This statement appears in paper[1] as theorem 3.2. It also appears in the introduction of a paper[2]. But the proof was omitted.
[1]Stover, Christopher. “A Survey of Quaternionic Analysis.” (2014).
[2]Gentili, Graziano and Daniele C. Struppa. “A New Theory of Regular Functions of a Quaternionic Variable.” Advances in Mathematics 216 (2007): 279-301.