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 $c>0$.