Sums of convex functions strictly convex in one variable

Question

Let $f_i:\mathbb{R}^n\to\mathbb{R}$, $i=1,2,\ldots,n$ be twice continuously differentiable, convex functions in $x = (x_1,x_2,\ldots,x_n)$. Let each $f_i$ be strictly convex in $x_i$. Is the function $$g(x) = \sum_{i=1}^n f_i(x)$$ strictly convex in $x$?

Answer 1

No. Consider $f_1 = f_2 = x^2+2xy+y^2.$