If $a+b+c+d=0$ and $\{a,b,c,d\}\subset[-1,1]$ so $\sum\limits_{cyc}\sqrt{1+a+b^2}\geq4$ Let $\{a,b,c,d\}\subset[-1,1]$ such that $a+b+c+d=0$. Prove that:
$$\sqrt{1+a+b^2}+\sqrt{1+b+c^2}+\sqrt{1+c+d^2}+\sqrt{1+d+a^2}\geq4$$
I tried Holder and more, but without success. 

Some thoughts
(For 3-variable problem, see: Prove $\sum_{\mathrm{cyc}} \sqrt{34x^2 + 28y^2 + 7z^2 - xy - 28yz + 41zx} \ge 9x + 9y + 9z$)
It suffices to prove the following:
Problem 1: Let $a, b, c, d \ge -1$ with $a+b+c+d = 0$. Prove that
$$\sqrt{1 + a + b^2} + \sqrt{1 + b + c^2} + \sqrt{1 + c + d^2} + \sqrt{1 + d + a^2} \ge 4.$$
(Thank @tthnew for verifying it by his routine.)
Problem 1 is equivalent to Problem 2 and Problem 3 below.
Problem 2: Let $x, y, z, w \ge 0$ with $x + y + z + w = 4$. Prove that
$$\sqrt{x + (y-1)^2} + \sqrt{y + (z-1)^2} + \sqrt{z + (w-1)^2} + \sqrt{w + (x-1)^2} \ge 4.$$
Problem 3: Let $x, y, z, w \ge 0$. Prove that
$$\sum_{\mathrm{cyc}} \sqrt{5x^2 + 9y^2 + z^2 + w^2 - 2xy + 6xz + 6xw - 6yz - 6yw + 2zw} \ge 4(x+y+z+w).$$
