Sectional Curvature of Convex Linear Combination of Metrics

Question

Suppose $g_1$ and $g_2$ are Riemannian metrics on a (say closed) manifold $M$. Denote $$|K_g| = \sup_{p \in M}\sup_{\Pi \subset T_pM}|K_p(\Pi)|,$$ where $K_p(\Pi)$ is the sectional curvature of the 2-plane $\Pi$. Then how is $|K_{\frac{1}{2}(g_1 + g_2)}|$ related to $(|K_{g_1}|, |K_{g_2}|)$? In particular, is there a universal constant $C$ (that depends only on $M$) such that $$|K_{\frac{1}{2}(g_1 + g_2)}| \le C \cdot \max\{|K_{g_1}|,|K_{g_2}|\}?$$