A uniform bound in the dimensional Brunn--Minkowski inequality for even log-concave measures

For every $n\ge 2$, we prove that there exists an exponent $p_n$ such that, for every even log-concave probability measure $\mu$ on ${\mathbb{R}}^n$, all nonempty symmetric convex sets $K,L\subseteq {\mathbb{R}}^n$, and all $\lambda \in [0,1]$, $$\mu (\lambda K+(1-\lambda )L{)}^{p_n}\ge \lambda \mu (K{)}^{p_n}+(1-\lambda )\mu (L{)}^{p_n},$$ where $$p_n\ge \frac{c}{n^2\ln n}$$ for some absolute constant $c>0$.