Complementary norm inequality for arithmetic and geometric means

Let $A$ and $B$ be positive definite matrices of order $n$. Define the weighted geometric mean at $t=1/2$ by

$$A\sharp B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$

and define

$$A\natural\natural B=A^{1/2}(B^{1/2}A^{-1}B^{1/2})^{1/2}A^{1/2}.$$

Let $\lVert\cdot\rVert_p$ denote the Schatten $p$-norm.

Complementary norm conjecture. For $1\leq p\leq\infty$,

$$\lVert A+B+A\sharp B+A\natural\natural B\rVert_p\leq\lVert A+B+2(A\natural\natural B)\rVert_p.$$

The paper proves the cases $p=1$ and $p=2$ and presents the inequality as a broader conjecture. The supplied context records that a related earlier conjecture for arbitrary unitarily invariant norms was proved for Schatten norms, but gives no resolution of this complementary inequality.