Uniqueness of the maximizer in $\sup_{X} E[|X+Z|^m]$ where $X$ is two-valued and $E[|X| ]\le c$ This question is a follow up to a previous question. 

Find
  \begin{align}
&\sup_{X}  E[|X+Z|^m] \quad (*)\\
&\text{ s.t. }  X \text{ has two mass points}, E[|X|^k]=c, X \text{ is indpendent of } Z
\end{align}
  where $Z$ is standard normal. Also for obvious reasons we assume that $0< m \le k$. 

In the original question, it was assumed that the sup in $(*)$ is achievable. However, it is not the case and there are case when the sup is not be attainable. 
Another, question that is interesting is to show that if the $\sup$ is achievable it is achieved by a unique distribution which is the question of this post.   

Question:  If the $\sup$ in $(*)$ is achievable is the maximizer unique?

Comment: not that if $m=k=2$ then 
\begin{align}
\sup_X E[|X+Z|^2]= \sup_X E[|X|^2] +E[|Z|^2]= c+1
\end{align}
and is attain by all two-distribution with $E[|X|^k]=c$. So, clearly it is not unique.  However, I think this case is the  exception.  
 A: Let $g(x) = E|Z+x|^m$ and let $\Gamma\subseteq\mathbb R^2 = \{(t^k,g(t)):t\ge 0\}$ be the graph of $g(x^{1/k})$.  Let $S$ be the convex hull of $\Gamma$ and let $\overline S$ be the closure of $S$. Points in $S$ are representable as $(E|X|^k, E|Z+X|^m)$ for some probability distribution on $\mathbb R_+$, and points in $\overline S$ are representable as limits of such points.  Let $\gamma^*$ be the upper concave envelope of $\Gamma$, that is, the pointwise infimum of the set of concave functions $f$ such that $f(x)\ge y$ for all $(x,y)\in\Gamma$. Let $p=(c,\gamma^*(c))$. The solution to the OP's supremum problem is $\gamma^*(c)$. Three cases are distinguishable.  Case 1:  $p\in \Gamma$.  In this case, the supremum is attained by a point mass at $c^{1/k}$.  (This includes the OP's $k=m=2$ example.) Case 2: $p\in S{\setminus}\Gamma$, in which case the supremum is attained by a mixture of two point masses but not by a single point mass.  Case 3: $p\in\overline S{\setminus }S$, in which case the supremum is not attained but is only approximated by mixtures of two point masses.
All three cases can occur, depending on the values of $c$, $k$, and $m$.  If  $\gamma^*$ is strictly concave at $c$ (that is, $\gamma^*{''}(c)<0$ ) then $p\in\Gamma$ and Case 1 obtains, and $p$ is an extreme point of $\overline S,$  and the representing measure is unique. In Case 2, $p$ is not an extreme point of $\overline S$ but rather lies on a finite line-segment facet of $\overline S$.  In this case $g(x^{1/k})$ is convex in some neighborhood of $c$. The representing measure is again unique, being supported by the pair of point masses representing the endpoints  of the facet containing $p$. Case 3 obtains when $c>0$ and $k=m=1$, in which case $g$ is convex and $\gamma^*(x) = x+g(0)$.  Here $p$ lies on part of the boundary of $\overline S$ containing a half-infinite line.
Numerical experiments (using the gsl package's  gsl_sf_hyperg_1F1 routine)  seem to show that Case 1 obtains whenever $k\gt 2$, and that Case 3 obtains when $1<m=k<2$.  When $k<2$ the function $g(x^{1/k})$ always seems to have a finite inflection point, and there seems to exist a     $c_0$ (that depends on $k$ and $m$) such that if $c<c_0$ Case 2 obtains but if $c>c_0$ then Case 1 obtains.
