Equivalence of positive corotational rates and strong monotonicity conditions

Let $\mathcal{E}: \operatorname{Sym}^{++}(3) \to \operatorname{Sym}(3)$ be a spatial strain tensor with strongly monotone scale function $e: \mathbb{R}^+ \to \mathbb{R}$. For a corotational rate $\frac{\mathrm{D}^{\circ}}{\mathrm{D} t}$, let $B\in\operatorname{Sym}^{++}(3)$ and $D\in\operatorname{Sym}(3)$, and write $\ell(t)>0$ for the one-dimensional stretch and $\overline D=\dot\ell(t)/\ell(t)$. Equivalence conjecture. The rate $\frac{\mathrm{D}^{\circ}}{\mathrm{D} t}$ is positive, meaning

$$\left\langle \frac{\mathrm{D}^{\circ}}{\mathrm{D} t}[B],D\right\rangle>0$$

for every nonzero $D\in\operatorname{Sym}(3)$, if and only if the corresponding one-dimensional and three-dimensional conditions hold:

$$\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}[e(\ell(t))]\dot\ell(t)>0\quad\text{for every }\dot\ell(t)\ne0,$$ $$\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}[\widehat e(\log\ell(t))]\overline D>0\quad\text{for every }\overline D=\frac{\dot\ell(t)}{\ell(t)}\ne0,$$

and

$$\left\langle\frac{\mathrm{D}^{\circ}}{\mathrm{D}t}[\mathcal{E}(B)],D\right\rangle>0\quad\text{for every }D\in\operatorname{Sym}(3)\setminus\{0\}.$$

The claim proposes a multidimensional characterization of positivity for corotational rates in terms of the corresponding one-dimensional strong-monotonicity inequalities and the strain tensor $\mathcal{E}$.