Complementary McCarthy inequality beyond the known parameter ranges

Let $A$ and $B$ be positive operators and let $q\in\mathbb{R}$. Complementary McCarthy conjecture.

$$\operatorname{Tr}(A+B)^q\leq \operatorname{Tr}A^q+\operatorname{Tr}B^q+(2^q-2)\operatorname{Tr}(A^{1/2}BA^{1/2})^{q/2}$$

for $-2<q<0$, while the reversed inequality should hold for $3\leq q$. The displayed ranges complement the theorem known for $q\leq-2$ and $1\leq q\leq2$; numerical simulations support the conjecture, but the source gives no resolution.