Is it true that the intersection of two dense subspace of a linear normed space is also dense provided that one of them is of finite codimension?

Question

Let $X$ be a normed vector space and $X_1,X_2$ be two dense subspace of $X,$ in Is is true that the intersection of two dense subspaces of a linear normed space is also dense? it was showed by an example that $X_1\cap X_2$ is not necessarily dense in $X,$ but in that example $\mathrm{codim}X_1=\mathrm{codim}X_2=+\infty.$ So I wonder if $\mathrm{codim}X_1$ or $\mathrm{codim}X_2$ is finite, will the intersection $X_1\cap X_2$ be dense in $X$ ?