Pythagorean-like inequality for isoperimetric profiles of product measures

Let $\mu_{1}$ and $\mu_{2}$ be log-concave probability measures on Riemannian manifolds $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$, respectively. For $a$ in the domain of the isoperimetric profiles, consider the product measure $\mu_{1}\otimes\mu_{2}$ and positive numbers $a_{1},a_{2}$ satisfying $a_{1}a_{2}=a$. Pythagorean-like inequality. One has

$$\left(\frac{I_{\mu_{1}\otimes\mu_{2}}(a)}{a}\right)^{2}\ge\inf_{a_{1}a_{2}=a}\left[\left(\frac{I_{\mu_{1}}(a_{1})}{a_{1}}\right)^{2}+\left(\frac{I_{\mu_{2}}(a_{2})}{a_{2}}\right)^{2}\right],\qquad\forall a.$$

The conjecture would remove the finite-$L$ assumption from the paper's equivalence theorem and is motivated by the corresponding inequalities obtained from Bobkov's arguments. Its validity is not established in the supplied text.