Not a bounty candidate. Just a pictorial comment.
Make an isoline/contour plot in the $(a,b)$-plane of the function:
$$
f(a,b) = a^{|b-a|}+b^{|c-b|}+c^{|a-c|} - \frac52 \quad \mbox{with} \quad c=3-a-b
$$
Then this is what we get. The blue spots are where $\,|f(a,b)| < 0.02$ . There seem to be several of these minimum values.
I wish the rigorous proof producers among us good luck.