Let $\ell_{p_1,p_2}=(\mathbb{R}^{m\times n},\|\cdot\|_{p_1,p_2})$ be the space of $m\times n$ matrices endowed with the mixed-norm

$$ \|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m |x_{ij}|^{p_1}\right)^{p_2/p_1} \right)^{1/p_2}. $$

On the other hand, given a normed space $(\mathbf{E},\|\cdot\|)$, its modulus of convexity is defined as the function $\delta:[0,2]\to[0,1]$

$$ \delta(\varepsilon) = \inf \left\{ 1-\left\|\frac{x-y}{2}\right\|: \,\, x,y\in\partial B(0,1),\,\,\|x-y\|\geq \varepsilon\right\}, $$ and we say its modulus of convexity is of power type $q\geq 2$ if there exists $c>0$ such that $\delta(\varepsilon)\geq c \varepsilon^q$.

In the study of the modulus of convexity of $\ell_p$ spaces, there is an interesting behavior of the power type, which turns out to be $q=\max\{2,p\}$. The case $p\geq 2$ can be derived from Clarkson's inequality; and for the case $1\leq p <2$ see e.g., here. I am interested in how the modulus of convexity behaves for mixed-norm spaces $\ell_{p_1,p_2}$. The cases I know are the following:

In the case $p_1,p_2\geq 2$, I believe this paper gives a tight Clarkson-type inequality, which can lead to the right power type for the modulus of convexity (if I am not mistaken, the power type is $q=\max\{p_1,p_2\}$).

It is also known that when $1\leq p_1,p_2\leq 2$, then the power type is 2 (see, e.g. this paper, where it is stated somewhat differently).

Finally, my question is:
*What is the power type of the modulus of convexity for the space $\ell_{p_1,p_2}$ when $p_1<2$, $p_2>2$ (and the reverse case, if it makes any difference)?*