Kernels of seminorms

Question

Suppose $p_1,\cdots, p_n: X \to \mathbb{R}$ are seminorms defined on a vector space $X$, and $q$ is another non-zero seminorm such that $$\{0\}\neq \bigcap_{i=1}^n \text{Ker}(p_i)\subset \text{Ker}(q).$$ Can we conclude $q\leq c_1 p_1 +\cdots c_n p_n$, for some reals $c_1, \cdots, c_n \geq 0$? Here $p_i$'s are linearly independent and $c_i$'s are not all zeros. Are there counterexamples?