Say $X$ is a smooth algebraic variety, $U$ is a Zariski open set in $X$, $L$ is a local system on $U$, and $IC(L)$ the intersection cohomology sheaf on $X$ which restricts to $L$ on $U$. Then is:

$$\dim L_u \ge \sum_i \dim \mathrm{H}^i(\mathrm{IC}(L)_x) $$

for $u \in U$ and all $x \in X$?

If not, is it true after e.g. putting in some scalar depending only on the dimension of $X$?