Sharp Poisson transform inequality on the Heisenberg group

Let $n$ be a positive integer, let $a$ satisfy $-2n<a<0$, and let $P_a$ be the Poisson transform from $H^{2n-1}$ to $H^{2n+1}$. Set

$$p=\frac{4n}{2n+a},\qquad q=\frac{4n+4}{2n+a}.$$

Sharpness and optimizer conjecture. There exists a sharp constant $C>0$ such that

$$\|P_af\|_{L^q(H^{2n+1})}\leq C\|f\|_{L^p(H^{2n-1})},$$

and the optimizers are translations, dilations, and multiples of the function

$$f(z',t')=((1+|z'|^2)^2+t'^2)^{-\frac{a+2n}{4}}.$$

This conjecture asks for the sharp endpoint inequality associated with the Poisson transform in the Heisenberg setting and a classification of all optimizers. The surrounding discussion states that the corresponding result is known in the real case, while its Heisenberg analogue is open.