  We show that for any log-concave measure $\mu$ on $\mathbb{R}^n$, any pair of
symmetric convex sets $K$ and $L$, and any $\lambda\in [0,1],$
$$\mu((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda)
\mu(K)^{c_n}+\lambda\mu(L)^{c_n},$$ where $c_n\geq n^{-4-o(1)}.$ This
constitutes progress towards the dimensional Brunn-Minkowski conjecture (see
Gardner, Zvavitch \cite{GZ}, Colesanti, L, Marsiglietti \cite{CLM}). Moreover,
our bound improves for various special classes of log-concave measures.
