A dimension-independent strict submultiplicativity for the transposition map in diamond norm

We prove that there exists an absolute constant $\alpha <1$ such that for every finite dimension $d$ and every quantum channel $T$ on $\mathsf{L}({\mathbb{C}}^d)$, ${\parallel \mathrm{\Theta }\circ (\mathrm{i}\mathrm{d}-T)\parallel }_{\diamond }\le \alpha {\parallel \mathrm{\Theta }\parallel }_{\diamond }{\parallel \mathrm{i}\mathrm{d}-T\parallel }_{\diamond }$, where $\mathrm{\Theta }$ is the transposition map. In fact we show the explicit choice $\alpha =1\mathrm{/}\sqrt{2}$ works.